# All Questions

**1**

vote

**0**answers

3 views

### Outer automorphisms of direct product of residually finite groups

Let $A,B$ be finite generated residually finite groups such that $\mathop{Out}(A)$ and $\mathop{Out}(B)$ are residually finite.
Is $\mathop{Out}(A\times B)$ residually finite?
If not, what is the ...

**0**

votes

**0**answers

2 views

### Probability of having a connected network in a random graph

I'm trying to solve this programming problem out of interest in my spare time and want to make sure my maths is correct.
"The people of Absurdistan discovered how to build roads only last year. After ...

**0**

votes

**1**answer

64 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**8**

votes

**2**answers

303 views

### Finite etale atlas for Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.
Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?
What if $X$ is an algebraic space ...

**0**

votes

**1**answer

31 views

### locally topologically finitely generated t.d.l.c. group

I am trying to find an example of the following situation.
$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is ...

**5**

votes

**1**answer

84 views

### Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...

**0**

votes

**1**answer

78 views

### subspace in pseudotopological space

Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...

**3**

votes

**1**answer

55 views

### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$):
$$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$
where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...

**0**

votes

**1**answer

2k views

### Region and domains? [closed]

Is every region a domain? Am I correct that I understood the definition of domain to be an open, connected set? Does every region have to be a domain? For example:
$|z-1+i|\le 3$ is a region if I've ...

**6**

votes

**10**answers

800 views

### Examples of $G_\delta$ sets

Recall that a subset A of a metric space X is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are three ...

**1**

vote

**0**answers

30 views

### Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back.
Let $GH^*(X,Z)$ be the subgroup of ...

**6**

votes

**2**answers

250 views

### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function
$P_i = px^i + qy^i + rz^i$
where $x,y,z$ are coordinates. I have a few related ...

**2**

votes

**0**answers

58 views

### Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...

**2**

votes

**1**answer

79 views

### Canonical divisor for conic bundles

Is there a general formula for the canonical divisor $K_X$ of a smooth conic bundle $X$?
Motivation: for smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $K_X = \mathcal{O}_X(d-n-1)$. But ...

**0**

votes

**0**answers

42 views

### Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$
conjecture with extra gcd conditions (observe that this
is different from the $abc$ and the $abcd$ conjectures).
This thesis p. 20 defines the ...

**2**

votes

**0**answers

22 views

### Optimal planar net for catching convex shapes

Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...

**-4**

votes

**0**answers

61 views

### I am doing P.H.D. in İstanbul University. My research is about Character Theory of Finite groups [on hold]

How can I define the p(t)-adic valuation and absolute value on F(t)?

**11**

votes

**2**answers

234 views

### How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...

**0**

votes

**0**answers

25 views

### Algebra Constructions

What kind of constructions of associative algebras are normally used in theory and examples besides the following ones:
group algebra
monoid algebra
tensor product
direct sums and products
...

**6**

votes

**2**answers

290 views

### Isotrivial fibrations over $\mathbb P^1$

First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard.
$S$ is a smooth complex projective surface with a fibration $f$ over ...

**8**

votes

**1**answer

204 views

### Existence of certain “nondegenerate” function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...

**-1**

votes

**0**answers

42 views

### Category of compact manifold [on hold]

Question 1: What we know about the sub-category $Met$ of $Top$ whose objects are metrizable topological spaces and whose morphisms are the isometries.
Question 2: Let $X$ be any compact manifold. We ...

**0**

votes

**0**answers

45 views

### A curious property of Ramanujan's function $\tau(n)$

As it is well known, Ramanujan's $\tau(n)$ function can be defined through the power series expansion of the modular discriminant:
$$\Delta(q)=q\prod\limits_{n=1}^\infty (1-q^n)^{24}=\sum ...

**3**

votes

**2**answers

55 views

### Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?

**0**

votes

**1**answer

61 views

### Sylow-subgroups of the group of units of a finite field

Let $K$ be a finite field with $p^n$ elements. Then the group of unit is cyclic of order $p^n-1$. What are the primes which are dividing $p^n-1$? What are the Sylow-subgroups of the group of units of ...

**-3**

votes

**0**answers

71 views

### Finding closed form of : $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

$ \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\dd}{{\rm d}}$
Finding closed form of the below:
$$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
This ...

**6**

votes

**1**answer

733 views

### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...

**-4**

votes

**0**answers

42 views

### Legendre symbol problem [on hold]

Let $p,q$ different odd prime numbers and $a$ a positive odd number, Prove that :
$$p \equiv - q\ \ \ \ (mod \ \ 4a )\Longrightarrow \Bigg(\frac{a}{p}\Bigg) =\Bigg(\frac{a}{q}\Bigg) $$
Where ...

**2**

votes

**1**answer

114 views

### Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...

**3**

votes

**1**answer

70 views

### expression for infinite series with powers of factorial in denominator

The series
$$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$
has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...

**4**

votes

**1**answer

200 views

### Sum of a random number of identically distributed but dependent random variables?

Background
Let $X_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1]
$$
\begin{align*}
...

**-2**

votes

**0**answers

41 views

### How to show this Legendre Symbol Problem [on hold]

Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$,
Prove that:
$\exists x, y \in \mathbb{Z};\,\, \gcd(x,y) = 1$ such as $x^{2} +ny^{2} \equiv 0\, (\mod p) \iff ...

**0**

votes

**0**answers

21 views

### Variance of sums of correlated variables when sampling without replacement?

Background
Let $X_t$ be an alternating renewal process, a stochastic process on the state space {0, 1} where 0 = 'Broken' and 1 = 'Working'. Let $U$ be the cumulative sojourn working during an ...

**-1**

votes

**1**answer

36 views

### extension of a continuous function [on hold]

Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$?
...

**13**

votes

**0**answers

110 views

### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**5**

votes

**1**answer

496 views

### Publishing an elementary proof of a less-general and less-useful version of a classic result?

Background
Let $X_t$ be a stochastic process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's uptime). It is well-known ...

**2**

votes

**1**answer

276 views

### Measure concentration for law of large numbers

The classical law of large numbers states that
$$\frac1k\sum_{i=1}^k X_i \rightarrow \mathbb{E} X_1$$
for i.i.d. $X_1, X_2, \ldots$ with finite $L^1$ norm.
I was wondering whether is it possible to ...

**2**

votes

**0**answers

51 views

### Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...

**-2**

votes

**0**answers

17 views

### Formula for unequal share distribution [on hold]

What formula would I use to distribute $M$ shares among $N$ shareholders, such that shareholder $X_i$ has 3/2 as many shares as shareholder $X_{i+1}$?
P.S.
I apologize if the tag isn't relevant. I ...

**11**

votes

**1**answer

740 views

### The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13

When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...

**0**

votes

**0**answers

121 views

### Problem regarding orthogonal vectors

Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...

**9**

votes

**0**answers

220 views

### Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...

**2**

votes

**1**answer

117 views

### The definition of computational complexity or complexity measure of computing reals [on hold]

A real $r$ is computable,if for any $i\in \mathbb{N}$,the $i$ bits can be outputed by a Turing Machine or an algorithm. So, what is computational complexity or complexity measure of computing the ...

**-5**

votes

**0**answers

49 views

### Exactly 2 Girls - Conditional Probability [on hold]

This is very confusing to me. I am really new with this stuff.
A couple wants to have 3 or 4 children, including exactly 2 girls. Is it more likely that they will get their wish with 3 children or ...

**6**

votes

**1**answer

209 views

### Bounds on the derivative of a Riemann map

Let U be a simply connected domain with smooth boundary in the complex plane, and let $\mathbb D$ be the unit disc. Is there a nice sufficient condition for the existence of a biholomorphic map ...

**-5**

votes

**0**answers

37 views

### Integral of (2-x)/(x-1) Really stumped [on hold]

So I tried doing this:
I have integral (2-x)/(x-1)
I used a substitution ;
u = x-1
x= u+1
du = dx
So then (2-u-1)/u du
then : 1/u - 1
Then I integrate and get
ln u - u
But when I plug ...

**10**

votes

**1**answer

93 views

### Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...

**0**

votes

**2**answers

2k views

### Minimal normal subgroups of a finite group

I have encountered a few problems regarding the minimal subgroups of a finite group $G$. Any references and/or answers regarding the following questions will be very welcome.
1)If $G$ is a finite ...

**5**

votes

**0**answers

89 views

### A question about three forcings

Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 ...

**3**

votes

**0**answers

49 views

### The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now ...