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multivariable calculus – Expression for potential vector of a central field

I know that for the central field
$$
{bf F(x)}=F({bf x})frac{bf x}{|{bf x}|^{3}}=F(x_{1},x_{2},x_{3})left(frac{x_{1}}{|{bf x}|^{3}},frac{x_{2}}{|{bf x}|^{3}},frac{x_{3}}{|{bf x}|^{3}}right)
$$

holds $nablacdot{bf F(x)}=delta_{0}$, where $|bf x|$ is the euclidian norm and $delta_{0}$ is the Dirac delta. If one consider a region of the space that doesn’t surround the origin, then $bf F(x)$ should have a potential vector $bf A$ such that $nabla times {bf A}={bf F}$, because in this case we can safely say that $nablacdot{bf F(x)}=0$.
But is there an analytical expression for that potential vector? Moreover, my reasonings are correct or am I wrong?

nt.number theory – Central simple algebras via cohomology

I am following the book Central Simple Algebras and Galois Cohomology, by Gille and Szamuely (I am using the second edition). In section 2.4, the authors remark that the tensor product induces a natural map $mathrm{CSA}_K(m)times mathrm{CSA}_K(n)to mathrm{CSA}_K(mn)$. Since we have previously identified degree $n$ central simple algebras split by $K$ with cohomology classes in $mathrm{H}^1(mathrm{Gal}(K/k),mathrm{PGL}_n(K))$, there is a corresponding map
$$mathrm{H}^1(mathrm{Gal}(K/k),mathrm{PGL}_m(K))times mathrm{H}^1(mathrm{Gal}(K/k),mathrm{PGL}_n(K))to mathrm{H}^1(mathrm{Gal}(K/k),mathrm{PGL}_{mn}(K))$$

The authors claim that this corresponding map is the one induced by the natural map $mathrm{GL}_m(K)times mathrm{GL}_n(K)tomathrm{GL}_{mn}(K)$. I don’t see why this is true. Any help?

Is there a package to download data from the Peruvian central bank in R?

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Reference requirement – central limit theorem versus entropy in a dynamic system context

A dynamic system $ (S ^ 1, T, mu) $, $ T_ * mu = mu $, $ T $ ergodic, $ S ^ 1 $ is circle. Suppose it has a central limit theorem.

Would you like to know the relationship between its measurement-theoretical entropy $ h _ { mu} (T) $ and the central limit theorem? Is there a good reference for that? Thank you so much!

For theoretical entropy $ h _ { mu} (T) $See https://en.wikipedia.org/wiki/Measure-preserving_dynamical_system#Measure-theoretic_entropy

For the central limit theorem means: for every smooth observable $ phi $ on $ S ^ 1 $ With $ int phi d mu = 0 $, $ frac { sum_ {i le n} phi circ T ^ i} { sqrt {n}} stackrel {d} { longrightarrow} N (0,1) $.

c ++ – publisher-subscriber architecture with central registration

This is my first question on this forum, so I hope that I follow the rules correctly.

I was recently assigned to a project and architecture is very important. I would therefore like to inquire with you in order to obtain an external and critical opinion.

Publisher-subscriber architecture with a central registration in which agents can either promote their functions or search for a specific function.

The project must be developed with C / C ++. A GUI used to create an agent chain is a nice thing (not necessarily C or C ++).

  • Set up publisher subscribers with ZeroMQ
  • The central registration role (or bulletin board) is only to connect agents between them using sockets
  • Data serialization is carried out with either MessagePack or FlatBuffers

It is the first time that I have been asked to develop a publisher / subscriber architecture and I have never used the libraries listed above.

Does my approach look good to you? Have you ever used these libraries? Do you recommend them or others?

Thank you for your input!

Probability: Central limit for globally dependent binary random variables

To let $ X in {0,1 } ^ n $ be a multivariate binary random vector with first moments $ mu in (0,1) ^ n $ and central moments of higher order
begin {align}
sigma_ {ij} & = mathbb {E} ((X_i- mu_i) (X_j- mu_j)), quad i <j, \
sigma_ {ijk} & = mathbb {E} ((X_i- mu_i) (X_j- mu_j) (X_k- mu_k)), quad i <j <k,
end {align}

and so on. Suppose that $ sigma_ {ij} neq0 $ for all $ i neq j $i.e. that all variables are correlated. Note that the elements of $ X $ are not weakly dependent– Elements that are far away have non-zero covariances.

Are there limits? $ sigma_ {ij}, sigma_ {ijk}, dots $, in terms of $ n $ which are necessary for the distribution function of the random variables $$ S_n = n ^ {- 1} sum_ {i = 1} ^ n frac {(X_i- mu_i)} { sqrt { mu_i (1- mu_i)}} $$ converge to the standard Gaussian distribution?

Laptop – Dell Webcam Central Catch 22

Latitude E6300, I finally got the webcam from Device Manager to install the Dell Webcam Central software now to confirm that the camera is working. I assumed it would work on another laptop after another installation, but it didn't. So I want to do it right.

I download the DellWebcamSW and all searches, and I mean a lot of searches. This is the Dell Webcam Central software. When I start the installation process I get the error "Dell Webcam Central could not be found on your system".

This is supposed to be this software !!!!

This is supposed to be the installation for it, not an update. It is for version 2.0. I ran it in compatibility mode, as an administrator I'm not happy.

I have after many repeated attempts on the other laptop referred to earlier and after a dozen times with the same results and so many files that are said to have been downloaded with the same name AND I thought I had an overview of the that worked, but obviously got lost in the shuffle.

There must be something I missed. I even tried an older version 1.0, so I thought: if I look back at the file after downloading it, it has the same version number: 1.1.4.0. I could go on, but I know there has to be a better way to do it.

Linux route between two networks via several central routers

I have two networks and a handful of computers.
I want to fix the central error point.

Two networks:

  • network Foo:: 172.16.1.0/24
  • network Bar:: 172.16.2.0/24

Five servers on Linux (Ubuntu):

  • server A
  • server B
  • server C
  • server Y
  • server Z

All servers have a network interface: eth0.
Except server C, which has two interfaces: eth0 **AND** eth1.
server C is the only server connected to both networks and is the central router between the two networks. It has a dedicated interface for each network and enables IPv4 forwarding.

Here is a diagram:
Diagram 1

^^ this seems to work just fine, but I want some kind of failover or load balancing of traffic by adding another central router.

If it were possible to add another Linux server, we say servers D, also with two network interfaces to perform the same function as the server C?

I want servers D being a failover or somehow balancing the traffic and figuring out that part is difficult for me.
I imagine that I can add a secondary route to the same network and add a metric or somehow weight the route on a node.

Here is a diagram that I imagine the solution might look like:

Diagram 2

Would it be possible to configure this to either balance the traffic between servers C and D, or would there be some kind of "failover" option?

Here are some things that interest me, but I'm not highly skilled in network scenarios, and I'm not sure which would be the most appropriate solution:

  • ip route add via metric preference
  • OSPF
  • BGP
  • or iBGP or eBGP (what is the difference?)
  • Multi-Path TCP
  • CARP or VRRP

I want to fix the central error point (Server C) and duplicate the role of this router. What would be the most practical approach?