## 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&#39;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 ​​either 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.