probability theory – Given random variables X and Z, can I (symmetrically) constuct a Y such that X, Y, Z is a Markov chain?

The following problem is giving me a bit of a headache:

Let $X, Z$ be a pair of random variables.
Under which conditions can I construct a random variable $Y$ such that
$X, Y, Z$ is a Markov chain? In order to avoid trivial solutions such as $X=Y$ I additionally require the construction to be symmetrical with respect to $X$ and $Z$.
That is, if I change $X$ and $Z$ when defining $Y$, I obtain the same random variable.

I am thinking of $Y$ of some sort of variable that captures the common “information” between $X$ and $Z$. Maybe it even holds that $I(X, Y) = I(Y, Z)$?

So far I could only find such $Y$ if I assume $X$ and $Y$ to be correlated standard normal distributions. An answer under any (non totally trivial) assumption is welcome 🙂

plotting – How to symmetrically color using ColorFunction in ArrayPlot?

I am trying to make and colour an ArrayPlot and since these data points are supposed to represent the phase of a system, I want to make the colouring symmetric such that for example $0$ and $2pi$ are represented by the same colour since those points are equivalent. In general, I want the data points $epsilon$ and $2pi-epsilon$ to be represented by the same colour.

ArrayPlot[RandomReal[2 Pi, {100, 100}], PlotLegends -> Automatic, ColorFunction -> "Rainbow"]

The output of the code I have above looks like this:

enter image description here

encryption – gpg stopped decrypting a symmetrically encrypted file

Just yesterday I decrypted this same file using a key that I have written down, but today every time I try the same key gpg returns:

gpg: decryption failed: Bad session key

I suspect that either I was typing something wrong every time I decrypted this file and didn’t notice or there’s something wrong with the characters that are being entered by my keyboard.

I used gpg -c <file_name>

Also, gpg says it is AES256.CFB encrypted data, although I don’t remember seeing this CFB anytime I decrypted something in this computer, although I might be mistaken, neither did I set this option when encrypting.

I am using Manjaro 20.2.1 and gpg 2.2.25 with libgcrypt 1.8.7

Can anyone help me?

ct.category theory – Can the effective topos be considered symmetrically monoidal?

in the

Example (e) for a monosymmetric closed category with NNO without infinite by-products?

User Zhen Lin indicates that the effective topos are locally Cartesian closed. In nLab we have that locally Cartesian with terminal object closed means Cartesian closed, and Hyland (in his original paper on the effective topos) indicates that there is such a terminal object and he calls it $ 1 $ as usual. Cartesian closed thus implies Cartesian monoidal implies symmetric monoidal. Is this argument okay? Do I miss something?

Also, since we can represent morphisms in a symmetric monoidal category as string diagrams (from Joyal and Street), does that mean that we can do this for the effective topos? I want to draw this!

If so, could someone help me there? My knowledge of all this is pretty tight and I have only successfully made this connection.

GR Group Theory – List of small, non-Abelian, symmetrically represented groups

To let $ F_n $ be the free group that was created by $ x_1, ldots, x_n $ and let it go $ S_n $ be the symmetric group on $ {1, cdots, n } $, To let $ w = x_ {i_1} ^ { pm1} cdots x_ {i_s} ^ { pm1} $ be a word and for everyone $ sigma in S_n $, define $ sigma (w) = x _ { sigma (i_1)} ^ { pm1} cdots x _ { sigma (i_s)} ^ { pm1} $, We consider groups of the form

$$ G_n (w) = langle x_1, ldots, x_n mid sigma (w), sigma in S_n rangle, $$
from where $ w $ is a given word in $ F_n $, Such groups are called symmetric. For example, this can be proved
$$ G_4 (x_1x_2 ^ 2x_3x_4 ^ {- 1}) = langle x_1, x_2, x_3, x_4 mid sigma (x_1x_2 ^ 2x_3x_4 ^ {- 1}), sigma in S_n rangle $$is a non-Abelian order group $ 96,

My question is given $ n $What is the smallest non-Abelian symmetrically presented group? Each list of examples of non-Abelian symmetrically presented groups is also greatly appreciated.

at.algebraische Topologie – Is the natural geometric realization of symmetrically simplified quantities homotopically correct?

Remember that the category of $ sigma set $ from symmetrical simple sentences is the category of Vorblätter on $ Sigma $, the category of finite nonempy sets and all functions. The recording $ v: Delta to Sigma $ transmits the Kan-Quillen model structure via a quill equivalency $ sset $, In this "canonical" model structure on $ sigma set $Not every object is cofibrant (the cofibrants are the ones where the $ Sigma_n $ Action on non-degeneracy $ n $-simplices is free). Nevertheless, Cisinski has shown that $ v _! $ preserves all weak equivalencies (in fact $ v _! $ is also a quillen equivalence for the Cisinski model structure $ sigma set $that have the same weak equivalences and the cofibrations are the monomorphisms).

The composition with the usual geometric realization results in a quill equivalency $ | v _! – |: sigma Set Top $, which calculates the correct homotopy type for all symmetric simplicial sentences.

The only downside is that $ | v _! – | $ is not the most economical geometric realization imaginable. For example, $ | v_![1]| $ has two 1 cells and i think is infinitely great.

A more economical and "natural" geometric realization can be obtained by bypassing $ sset $ a total of. That is, leave $ | – |: sigma set on top $ be induced by the functor $ Sigma to Top $, $[n] mapsto delta ^ n $, Then $ |[n] | = Delta ^ n $ with the obvious CW structure.

Question 1: is $ | – | $ a left-hand quiquel equivalency with respect to the canonical model structure?

It is clear that the Serre cofibrations and acyclic Serre cofibrations are produced by the image of the canonical cofibrations and canonical acyclic cofibrations. So, if the answer is yes, the model tree is displayed $ Top $ is even taken $ | – | $ from $ sigma set $,

Question 2: is $ | – | $ a quill equivalence with respect to the Cisinski model structure?

Question 3: does $ | – | $ receive weak equivalences between arbitrary objects?

An affirmative answer to Question 2 would, of course, be an affirmative answer to Question 3.