Why does longitude correspond to Frobenius in arithmetic topology and other strange phenomena?

I'm trying to address Morishita's book Nodes and Premiums to learn a little more about arithmetic topology, but some analogies surprise me. I know that correspondence has to be addressed with a grain of salt, but some parts are so fundamental that I would like to understand them better.

  1. In the table (3.3) on page 50 of his book, Morishita writes that longitude, called $ beta $, should correspond to a Frobenius lift and meridian, called $ alpha $, corresponds to a taming inertia generator (both as elements of the maximum taming quotient of the absolute Galois group of a local field). It calls "longitude" a path that bypasses a hole in the boundary of a tubular neighborhood of the node and "meridian" the edge of a disc that is a "cross section" of the tube. If the knot was the denouement, this neighborhood would be the complete torus $ S ^ 1 times D ^ 2 $: in that case, $ alpha = partial D ^ 2 $ and $ beta = S ^ 1 $. This analogy does not support my view that inertia acts as a monodromy, which is "turning around holes", but I tried to pursue it. Then (page 63, after Theorem 5.1), he describes the analog of decomposition groups for an uninterpreted node. K $: he says that this group should be generated by "a circulating loop" K $"which I think is just the image of $ alpha $. Then I'm completely lost, because I would expect Frobenius to generate decomposition groups in un-simplified situations …
  2. In chapter 11, an experimental analogy with the Iwasawa theory is suggested. Nevertheless, it seems to me that something is odd, because the typical Iwasawa theory concerns a very wild branch, whereas the same table (3.3) on page 50 seems to indicate that there is no wild topological inertia. So the Galois group of $ X_ infty / X_K $, or $ X_K $ is a node complement and $ X_ infty $ is a $ mathbb {Z} $-covering $ X_K $, in a sense, resembles an "infinitely thinly branched cover" (which has no arithmetic analog) rather than a $ mathbb {Z} _p $-extension. Am I missing something? In the same sense, he has a small parenthesis between p. 144 and p. 145 where he writes that "[Ensupposantqu'a[assumingthata[ensupposantqu'un[assumingthata$ mathbb {Z} _p $-extension be branched to a first only, and this first be totally ramified]is a hypothesis analogous to the case of knot. "Why is it so? Or the fact that one $ mathbb {Z} $-cover must be branched to a single node, and the fact that there can not be a small, unbranched layer below seems obvious to me (at least if the basic variety is not $ S ^ 3 $, other $ pi_1 (S ^ 3) = $ 0 should say there is no unassembled extension).

matrices – Frobenius normal form of a doubly stochastic matrix

Yes $ A in M_n ( mathbb {C}) $then $ A $ is called reducible if there is a permutation matrix $ P $ such as
$$
P ^ top A P =
begin {bmatrix}
A_ {11} & A_ {12} \
0 & A_ {22}
end {bmatrix}, $$

in which $ A_ {11} $ and $ A_ {22} $ are square matrices of at least one order. Yes $ A $ is not reducible, so $ A $ is called irreducible. Note that with this definition, each matrix one by one is irreducible. A matrix $ A $ is irreducible if and only if its Led graph or digraph is strongly connected.

He is known (see, for example, Brualdi and Ryser [Theorem 3.2.4; MR1130611]) that if $ A in M_n ( mathbb {C}) $then $ A $ is irreducible or there is a permutation matrix $ P $ such as
begin {equation}
P ^ top A P =
begin {bmatrix}
A_ {11} & cdots & A_ {1k} \
& ddots & vdots \
& & A_ {kk}
end {bmatrix}, tag {1} label {fnf}
end {equation}

in which the matrices $ A_ {11}, dots, A_ {kk} $ are square and irreducible matrices. The matrix in eqref {fnf} is called Frobenius or irreducible normal form (of $ A $) (FNF) and it's not unique. However, the blocks $ A_ {11}, dots, A_ {kk} $ are unique until the similarity of permutation.

A matrix $ A $ is non-negative ($ A $ 0 ge) if $ a_ {ij} ge $ 0, $ 1 the i, j the n $. A non-negative matrix is stochastic if $ Ae = e $ and stochastic doubling if, in addition, $ A ^ top e = e $ (that is to say., $ A $ has rows and columns equal to one).

In a 1965 article, Perfect and Mirsky [MR0175917] indicate, without proof, that $ A $ is a doubly stochastic matrix, so each FNF of $ A $ is of the form
$$
begin {bmatrix}
A_ {11} & & \
& ddots & \
& & A_ {kk}
end {bmatrix},
$$

that is, each doubly stochastic matrix is ‚Äč‚Äčeither irreducible or is permutationally similar to a direct sum of irreducible and doubly stochastic matrices. After giving the result Perfect State and Mirsky:

"This result is almost certainly known.
derives very easily from the definitions, we omit the details of the
evidence. "- p.38.

This result is easy to establish for matrices of less than or equal to four, but does not seem obvious or seems to flow from the definitions above.

Question 1: Is this result obvious or is there a simple proof of this fact?

Question 2: Does anyone know of a reference?

Distance induced by the Frobenius norm between two multivariate normals

There is a standard on covariance matrices defined by the Frobenious standard. http://mathworld.wolfram.com/FrobeniusNorm.html
Can this be used to define a valid distance between two multivariate Gaussian distributions? Something like $ d ^ {2} left ( mathcal {N} left ( mu_ {1}, Sigma_ {1} right), mathcal {N} left ( mu_ {2}, Sigma_ {2 } right) right) = || mu_ {1} – mu_ {2} || _ {F} ^ {2} + || Sigma_ {1} – Sigma_ {2} || _ {F} ^ {2} $. I am looking for references / articles.