mining theory – Can miners collude to bypass the penalty based revocation system of the lightning network?

It’s named “penalty transaction” by the specs :).

Regarding your core question, you are right but this also requires this miner to be able to heal this block chain with a censored (the penalty) transaction. It therefore assumes that a single miner has more hashpower than the rest of the network combined.

Thus it can be stated as such:

The Lightning Network core security assumption relies on the fact that nobody can “51% attack” the network.

The decentralization of mining is a huge deal for Lightning (and all other L2s i know of).

cv.complex variables – Global Theory of Holomorphic Functions

I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D subset mathbb{C}$. Recently, I was looking at the last chapter of Lars Ahlfors Complex Analysis book and I discovered striking similarities between my approach and that of Weierstrass.

First, lets start with a definition of holomorphy. Lets say that a complex valued function $f(z)$ is holomorphic at $z_0$ if $lim_{z to z_0}frac{f(z)-f(z_0)}{z-z_0}$ exists and is finite. So holomorphy is implicitly a local property that not only involves a single point $z_0$ but all of the points in a neighborhood containing $z_0$.

Now let us proceed and try to develop a global theory of holomorphic functions. The idea is to fix a point $z_0$ in the complex plane and look at the behavior of a particular holomorphic function $f(z)$ in a neighbored $B(z_0, r)$. It will become more clear why I take this approach. Now we want to extend the domain of holomorphy of this fixed function $f(z)$ defined initially in an open neighborhood of $z_0$. One way to do this is to use an equivalence relation.

We say that $f(z)$ R $g(z)$ if $f(z) = g(z)$ for all $z$ in any open set containing $z_0$. This is clearly an equivalence relation which partitions the set of holomorphic extensions of $f(z)$ near $z_0$. By choosing a representative $g$ of each equivalence class we see that there exists open sets $U_g$ (the domain of $g$) such that $f(z) = g(z)$ on $B(z_0, r) cap U_g$ and $g$ extends $f$ to $B(z_0, r) cup U_g$. Now let $U_{z_0} = cup_g U_g$ where the union is taken over all equivalence classes described above. Then $U_{z_0}$ is the largest open set where we can find a holomorphic extension of $f(z)$ at $z_0$. That is, there exists a global extension $G(z)$ of $f(z)$ at $z_0$ in the sense that $G$ is holomorphic and the domain of $G$ is $U_{z_0}$.

Questions and Comments:

  1. Is this formulation of analytic continuation correct and is it any different from Weierstrass’ approach?

  2. I cannot prove existence and holomorphy of $G$ but I suspect that follows from a simple relation between the representatives of each equivalence class and $G$.

  3. How to I pass from a fixed point $z_0$ to the entire domain of $f$? If $D$ is a countable union of open sets then I think its possible.

nt.number theory – Finite groups arising as Galois groups of maximal unramified extension of number fields

I was wondering if it is known for which number fields the maximal unramified (non-abelian) extension is of finite degree or do we know the finite groups that arise as the Galois groups of these finite degree maximal unramified extensions.

I have seen the trivial group and the group of two elements as these Galois groups but not beyond that.

Thanks in advance.

field theory – why $F(a^2) subset F(a)$?

I have some doubt in this post

Let $E$ be an extension field of $F$. If $a in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$.

let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$

Since $a^2 in F(a)$, We have $F(a^2) subset F(a)$, then $kle n$

I don’t understand why $F(a^2) subset F(a)$?

My thinking: Take $ a in mathbb{R} $, $a subset a^2 implies F(a) subset F(a^2)$

graph theory – Generating triangulations with given topology

I am looking for information about the problem of identifying the heaviest minimal subset $Fsubset E$ of the edgeset $E$ of a complete symmetric graph $G(V,E)$ with randomly weighted edges such that $Gsetminus F$ has a given topology, e.g. is planar, and a triangulation, i.e. every remaining edge is an edge of a 3-cycle.

An exemplary problem would be to find a triangulation of a complete graph, whose vertices resemble points on a torus and edge weigths equal to euclidean distance, that allows for a planar embedding.

Being able to calculate the lightest planar triangulation of graphs with arbitrarily weighted edges would e.g. yield a meaningful definition of the planar convex hull of such graphs and thus yield an initial tour that can be expanded to a lightest Hamilton cycle by successive integration of vertices.

ct.category theory – $Gamma: mathcal C to text{Fun}(mathcal Z, mathcal C)$ has a left adjoint iff $F in text{Fun}(mathcal Z, mathcal C)$ has a colimit

Let $mathcal Z$ be a small category and $mathcal C$ any category. We then consider the category $text{Fun}(mathcal Z, mathcal C)$ the category of functors from $mathcal Z$ to $mathcal C$ and the functor $Gamma: mathcal C to text{Fun}(mathcal Z, mathcal C)$ such that
$$
Gamma(C):
begin{pmatrix}
mathcal Z &longrightarrow &mathcal C\
Z & longmapsto & C\
f & longmapsto & 1_C
end{pmatrix}
$$

where $f: Z to Z’$ and $1_C$ is the identity morphism on $C$.

I am trying to show that $Gamma$ has a left adjoint if and only if the colimit of every functor $F: mathcal Z to mathcal C$. The “if” part is not too hard but I have some difficulties to show the “only if” part.

Here is my idea: Suppose that there is $Omega : text{Fun}(mathcal Z, mathcal C) to mathcal C$ such that
$$theta_{F, C}: text{Hom}_mathcal C(Omega(F), C)cong text{Nat}(F, Gamma(C)).$$
For each $alpha in text{Nat}(F, Gamma(C))$ we can associate a cocone, i.e. for $f: Z to Z’$ the following diagram commutes
$$
begin{array}{ccc} F(Z)&xrightarrow{F(f)} &F(Z’)\ searrow&&swarrow\
&Gamma(C)(Z) = C = Gamma(C)(Z’)& end{array}
$$

where the arrows $F(Z) to C$ and $F(Z’) to C$ are given by $alpha_Z$ and $alpha_{Z’}$ respectively. Because of the above isomorphism, we can associate to $alpha$ a unique morphism $theta^{-1}_{F, C}(alpha) =u: Omega(F) to C$ so $Omega$ is a good candidate to be the colimit of $F$. We just have to find a family of morphism $mu_Z: F(Z) to Omega(F)$ such that $alpha_Z = u circ mu_Z$. My guess is that this family of morphism is given by the unit of the adjunction $eta_F in text{Nat}(F, Gamma(Omega(F)))$ but I am not able to show that
$$alpha_Z = u circ (eta_F)_Z = theta^{-1}_{F, C}(alpha) circ (theta_{F, Omega(F)}(1_{Omega(F)}))_Z$$
where the last equality comes from the expression of $eta_F$ in term of $theta_{F, Omega(F)}$. Does it seem right ? Does anyone know how to conclude ?

probability theory – Linear transformations making independent features dependent

I read about features and some relevant topics recently. I ran into an easy but very advanced question:

Why can two independent features become dependent after applying a linear transformation?

I think this is false, because:

For $x_1$ and $x_2$, after linear transformation $y_1=ax_1+b$ and $y_2=cx_2+d$, $y_1$ and $y_2$ are independent.

gr.group theory – What does it matter if a group has a non-elementary hyperbolic quotient?

My adviser recently shared a problem with me that seeks to establish non-elementary* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on separability or omnipotence, and that these could be relevant for examining profinite rigidity of hyperbolic 3-manifolds. Unfortunately, I’m not fully read up on these topics.

In this recent paper, Behrstock, Hagen, Martin and Sisto also seek to make headway on the question of hyperbolic quotients for mapping class groups. They have a discussion in their introduction of the relevance of this question, mentioning again separability and omnipotence, profinite rigidity, and placing things in the context of the virtual Haken conjecture. Again, I’m a little ignorant of these topics and their history of this point. So my question:

Q: Can anyone explain with some detail (or point me to some nice references as to) why it’s relevant that mapping class groups have hyperbolic quotients? Or why it’s helpful that any group has such a quotient?


*A Gromov-hyperbolic group is non-elementary if it is not virtually cyclic, i.e. is infinite and not virtually $mathbb{Z}$.

complexity theory – For s set $Ssubseteq RE$, so call feature of language $S=emptyset$ vs. $S={emptyset}$

Assume that all languages are over the alphabet $Sigma$. What you have here is a bit of ambiguity in the meaning of $emptyset$ (recall that the emptyset is defined w.r.t a universal set, and here $emptyset$ is used w.r.t different universal sets). Indeed, $S = { emptyset}$ refers the set of languages containing only the empty language, that is, in this case, $emptysetsubseteq Sigma^*$. Also, $S = emptyset$ refers to the empty set w.r.t to the universal set of all languages, that is, in this case $emptysetsubseteq 2^{Sigma^*}$.

As you noted, if $S = emptyset$, then $L_S = { langle Mrangle: L(M)in emptyset} = emptyset in text{R}$. Now if $S = { emptyset}$, then $L_S = { langle Mrangle: L(M)in {emptyset}} = { langle Mrangle: L(M) = emptyset} = E_{TM}$ which is known to be in $text{coRE}setminus text{R}$.

gr.group theory – Integer matrices which are not a power

In a group $G$, an element $g$ is said to be primitive if there is no $h in G$ and integer $n >1$ such that $g = h^n$.

I was wondering, in the case $G$ is $SL_n(mathbb{Z})$ or $SP_{2n}(mathbb{Z})$, if there exists a criterium for primitiveness of matrices. I actually even struggle to find examples of primitives matrices in these groups.

In $SP_{2}(mathbb{Z})$, the matrix $pmatrix{1 & 1 \ 0 & 1}$ is primtive. (that can be shown by considering its action on $mathbb{H^2}$, for example)

But this does not generalize (easily at least) to higher dimension. For example, and quite surprisingly maybe
$$
pmatrix {1 & 1 & 0& 0\ 0 & 1 & 0 & 0\ 0 & 1 & -1 & 0\ 0 & 1& -1 & 1 }^3 = pmatrix{1 & 1 & 0& 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0& 0 & 1}
$$

Anyway, it seems like some things should be known, but it is very hard to find anything on google since primitive matrix usually means something else …
I would appreciate any input.