Non-trivial zeros of Davenport-Heilbronn function

From the book The Riemann Zeta-Function by Karatsuba we know that for Davenport-Heilbronn like function equations, there are many non-trivial zeros off the critical line. However I found this paper on arXiv which indicates that there are no non-trivial zeros off the critical line. Can anyone help to clarify what mistakes the authors have made in this paper?

Will Nakamoto consensus work if a nontrivial portion of miners/nodes are on the moon?

The Nakomoto consensus should work regardless of where the miners are located, as long as they are using the same network. The internet on earth, as it currently stands, is fully connected. So miners in China, Germany, and the United States are all contributing to the network where information is travelling across the internet.

It could be conceivable that different planets (or the moon) would have their own internet at some point, disconnected from that of earth’s, but that seems less likely than being connected via satellite transmission. As long as the networks of the moon and Mars are connected to the internet on earth, everything should still work normally.

Edit: Thanks to @PeterWuille in the comments for correcting this information. Here’s what was said:

“Nakamoto consensus requires that the block propagation time between miners is negligible compared to the inter-block time. If it isn’t, miners which are latency-wise “close” to each other will find more blocks that proportional to their hashrate share (simply because further out miners have a delay before being able to start working on a block), resulting in a (geographical) centralization pressure. When the latency exceeds the inter-block time (like it would be between Earth and Mars) the networks will split and fail to converge even.”

ag.algebraic geometry – On non-trivial components of Gushel-Mukai threefolds and line transform

I was wondering whether following statements are proved somewhere else or expected?

  1. Let $X$ and $X’$ be smooth Gushel-Mukai threefolds(special or ordinary). Then I assume that $X$ is general and the non-trivial components of derived categories are equivalent $mathcal{A}_Xsimeqmathcal{A}_{X’}$, then $X’$ is also general.

  2. Let $X$ and $X’$ be smooth General Gushel-Mukai threefolds such that $mathcal{A}_Xsimeqmathcal{A}_{X’}$ and $X$ is ordinary(resp. special), then $X’$ is also ordinary(resp.special).

The motivation comes from the duality conjecture for GM varieties and a conjecture by Kuznetsov-Perry on inverse of this conjecture. It seems that by very careful study on smoothness and study of singular points on moduli spaces of semistable sheaves on $M_G(2,1,5)$ on (special or ordinary)smooth GM threefolds and minimal Fano surface of conics $C_m(X)$ on (special or ordinary)smooth GM, one can obtain the above statements.

Second, in the literature of Fano varieties, the line transform is not only defined for ordinary Gushel-Mukai threefold(as treated in a paper of Debarre-Iliev-Manivel), but also for special GM threefold. But if the above statements are true, then it seems that one can conclude that a line transform on a general special Gushel-Mukai threefold is an automorphism, which looks a little bit strange to me. But it looks like this is not super strange because line and conic transform will produce the two dimensional fiber of period map for ordinary GM threefold, but for special GM, it seems that one expects Torelli statement.

nt.number theory – Non-trivial soluion to $sum^{n}_{i=1}sum^{n}_{j=1,jneq i}(x_{i})^{(x_j)}=(sum^{n}_{i=1}x_i)^{(sum^{n}_{i=1}x_i)}$

This problem is from Mathematica Stack Exchange: https://math.stackexchange.com/questions/3956135/non-trivial-solutions-for-a-group-of-equations, with isn’t payed much attention.

Define an equation $A_n$ like the following:
$$sum^{n}_{i=1}sum^{n}_{j=1,jneq i}(x_{i})^{(x_j)}=(sum^{n}_{i=1}x_i)^{(sum^{n}_{i=1}x_i)}$$
For example, $A_3$ looks like the following equation:
$${x_1}^{x_2}+{x_1}^{x_3}+{x_2}^{x_1}+{x_2}^{x_3}+{x_3}^{x_1}+{x_3}^{x_2}=({x_1}+{x_2}+{x_3})^{({x_1}+{x_2}+{x_3})}$$

Let’s assume that for every non-negetive solutions for $A_n$, $x_ileq x_{i+1}$ for every $1leq i<n$, then there are two distinct non-negetive solutions for $A_3$, one is ${x_1}=0,{x_2}=0,{x_3}=2$, and the other is ${x_1}=0,{x_2}=1,{x_3}=1$.

We call a solution for $A_n$ ‘non-trivial’ if $x_{n-1}neq0$. The only known non-trivial solution is ${x_1}=0,{x_2}=1,{x_3}=1$ for $A_3$. The problem is: are there any more non-trivial solutions for $A_n$?
If so, please give an example.

ct.category theory – Simple example of nontrivial simplicial localization

Does anyone has a simple example of a 1-category $mathcal{C}$ and a collection of morphisms W such that the infinity-categorical / simplicial localization $mathcal{C}left[W^{-1}right]$ is not a 1-category?

Of course there are obvious “big” examples like CW complexes / derived categories, I’m looking for a small example that I’ll be able to understand combinatorially.

Thanks!

algebraic topology – Show element of fundamental group is nontrivial

I’m learning cohomology, and I’d like to show the following:

Let $i:mathbb RP^1tomathbb RP^n$ be the usual embedding taking $(x_0,x_1)mapsto(x_0,x_1,0,dots,0)$, where $nge2$. Further, let $v:S^1tomathbb RP^1$ be the fibration $(x_0,x_1)mapsto(x_0,x_1)$. Show that $(icirc v)$ is a nontrivial element of $pi_1(mathbb RP^n,*)$.

I tried proof by contradiction: Suppose $icirc v$ is homotopic to the constant map $c$ which sends everything to $*$. Then these induce the same maps in cohomology: $v^*circ i^*=(icirc v)^*=c^*$. Recall that $H^q(mathbb RP^m;mathbb Z/2mathbb Z)=mathbb Z/2mathbb Z$ for all $0le qle m$. Let $Omega_nin H^1(mathbb RP^n;mathbb Z/2mathbb Z)$ be the nonzero element, and similarly define $Omega_1in H^1(mathbb RP^1;mathbb Z/2mathbb Z)$. Then I already know that $i^*(Omega_n)=Omega_1$. The Gysin sequence shows that $v^*(Omega_1)=0$.

I wanted to show that $c^*(Omega_1)$ is nonzero. But I feel like I don’t quite understand cohomology groups/rings enough. If we write $Omega_n=text{cls}~omega_n$, where $omega_nin Z^1(mathbb RP^n;2)$ is a homomorphism $C_1(mathbb RP^n)tomathbb Z/2mathbb Z$, then the goal is to show that $$omega_nc_#:C_1(S^1)to C_1(mathbb RP^n)tomathbb Z/2mathbb Z$$ is not a coboundary, but I haven’t been able to do this. After all, isn’t $omega_nc_#$ just the zero map?

This is Exercise 12.24 in Rotman, and includes the hint that $pi_1(mathbb RP^n,*)congmathbb Z/2mathbb Z$ and $mathbb RP^1approx S^1$. But I didn’t use either one.

graph theory – Non-trivial spanning trees from tours

If $G(V,E)$ is a connected graph with $n$ vertices and $n$ edges, it contains exactly one cycle $Csubseteq G$ with at least 3 edges.

If side denotes an edge $cin Ccap E$ and diagonal an edge $ d_{uv}:=lbrace u,vrbracesubset Ccap V, d_{uv}notin Ccap E$ then a non-trivial tree can be generated from $G$ by repeatedly replacing in $E$ the heaviest side with the lightest diagonal until $C$ contains exactly 3 edges and finally deleting from $E$ the heaviest side of that triangle.

Question:
if the above algorithm is applied to optimal tours of plnare euclidean TSP instances, what is the smallest upper bound on the factor by which the resulting tree’s edge sum is larger than that of the MST of the TSP instance?

group theory – How do I find a non-trivial 2D representation of $Z_2 subset $ SL(2, C)?

I am trying to find a non-trivial two dimensional representation of $Z_2 subset SL(2, C)$, but I am a little stuck.

Here is what I did so far:

For M $in$ SL(2, $Z_2$):
begin{bmatrix}a&b\c&dend{bmatrix}
with det M = ad – bc = 1 and a, b, c, d $in$ {0, 1}.

I also found that the order of SL(2, Z$_2$) is 6.

So putting the pieces together, I think that this means that I am looking for a representation of six 2 x 2 matrices that each have a determinant of +1, where my matrix elements are either 0 or 1.

I found the following 3 matrices that satisfy these conditions
begin{bmatrix}1&0\0&1end{bmatrix} begin{bmatrix}1&1\0&1end{bmatrix} begin{bmatrix}1&0\1&1end{bmatrix}

This is where I am stuck: I am unable to find 3 more matrices, only consisting of 0 and 1, that satisfy the condition that my determinant is 1. How do I proceed?

dg.differential geometry – Non-trivial $mathbb{R^3}rightarrowmathbb{R^3}$ maps with constant singular values

It can be proved that all $mathbb{R^2}rightarrowmathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like

$$
begin{align}
x’&=lambda_2 x -frac{1}{lambda_2}sqrt{frac{lambda_2^2-lambda_1^2}{lambda_3^2-lambda_2^2}}int_0^{z}sin f(xi),mathrm{d}xi \
y’&=lambda_2 y +frac{1}{lambda_2}sqrt{frac{lambda_2^2-lambda_1^2}{lambda_3^2-lambda_2^2}}int_0^{z}cos f(xi),mathrm{d}xi \
z’&=frac{lambda_1lambda_3}{lambda_2}z
end{align},
$$

for an arbitrary differentiable $f$ and $lambda_3>lambda_2>lambda_1$, whose differential has constant singular values $lambda_3,,lambda_2,,lambda_1$. So I wonder if one can classify or say something generic about such $mathbb{R^3}rightarrowmathbb{R^3}$ maps, as is the the case in two dimensions.

gr.group theory – Group with non-trivial center containing trivially-intersecting copies of itself

I’m trying to think of an example of a group $G$ with non-trivial center such that there exist subgroups $H_1,H_2le G$ both isomorphic to $G$ and satisfying $H_1cap H_2={1}$. Does such a group exist? Preferably an easy-to-state example that I’m just not thinking of? (Ideally finitely generated, but I won’t insist on it.)

If the center is trivial then this is easy (free groups work, I think any centerless RAAG will work, various Thompson-like groups work), but requiring non-trivial center seems to complicate things. I would also be happy with an example that requires intersecting more than two copies of $G$ inside $G$ to ensure a trivial intersection (as long as it’s finitely many copies).