Prove that the language $ {a ^ ib ^ i | i geq 0 } $ is not regular? (Let's just consider $ a ^ nb ^ n $, where $ n $ is the pump length?

I think to prove that $ {a ^ ib ^ i | i geq 0 } $ is not regular, we just have to consider the string $ a ^ nb ^ n $ (that's in the language) and apply the pumpable lemma. But I'm not sure how to deal with the pumpable lemma (although I know that it applies to the string we chose because the string is at least one string) $ n $ Long).

Spells – Can nerve jitter be used during a surprise round, let's say straight away if you find yourself being mugged?

So with all this information, if I should have one contingency spell (speed) with the condition: whenever I cast Nerveskitter: Activate.

What would happen? I would be able to act immediately, then the rest of the surprise round would happen (let's say I'm immune to drowsiness or I fought back) Quick recovery feat) what then?:

  • Do I start first and don't need an initiative?
  • I still have to roll for the initiative?
  • I think you don't have to take any initiative if you have contingency (quickness) on the condition that if a creature does something hostile to me: activate (no surprise round, let's say to simplify it), you go first and if you are not dazed, do you start first? Perhaps you still have to take the initiative in this case?

RAW answers if possible (FAQ, 3.X, I would normally also allow Pathfinder, but nerve jitter has been removed in Pathfinder)

(I know there is more than one question on this question. It was easier for me.)

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tls – Man-in-the-Middle Attack (ACME / Let's Encrypt) on Authorization Key?

Looking at https://letsencrypt.org/how-it-works/, I felt that a man-in-the-middle attack might be possible during the domain verification phase.

During this phase, the administrator is asked to Run the challenge of dropping a file on the web server and sending a signed nonce to the certification authority.
What happens if a man in the middle generates his own (authorization) key pair (while the administrator performs the domain check) and signs the nonce with the malicious key pair? The administrator would have performed the challenge by posting a file on the server, and the certification authority would have received a maliciously signed nonce (if the middle man is able to discard the signed nonce from the administrator). This would mean a compromised authorization key for the middle-man control of a domain.

The solution (in my opinion) would be to just publish the signed nonce on the website (in a path set by the certification authority) instead of sending it to the certification authority so that the "signature exchange" can not take place. However, this is not illustrated in the explanation "How it works" by Let's Encrypted.

Do I understand something wrong or how is such an attack prevented?

Numeric Integration – A list was obtained from external software (from COMSOL – to Mathematica). Let's make it a continuous function

I solved an equation with COMSOL. The solution in the form of a nested list with the term {x,y,z} is here.
These

lst = Flatten(
   Import(NotebookDirectory() <> "trial.xlsx") /. {x_, y_, z_} /; 
      Abs(z) < 10^-6 -> {x, y, 0}, 1);

Imports the list if it is in the same directory as the workbook. This is how it looks like:

ListPlot3D(lst, ColorFunction -> "Rainbow")

Enter image description here

My goal now is to build a sufficiently smooth function based on this list that I could use for further post-processing. In particular, I have to do a pretty special integration.

I'll try the following idea: let's define a function, g(x,y) so for every point {x,y} it takes the value z of the point whose coordinates are closest to the point in the XY plane {x,y} in this plane.

I tried the following. Here is the list of XY coordinates of the points of the original list, lst;

lstXY = Transpose(Transpose(lst)((1 ;; 2)))  

With this list I do the relevant function:

g(X_, Y_) :=lst((Position(lst, Join(Flatten(Nearest(lstXY, {X, Y})), {a_}))((1,1)))) // Last

You can check if it actually returns a value:

g(-60, 50)

(*  0.123899 *)

You can look at the plot:

    Plot3D(g(X, Y), {X, -65, 139}, {Y, 0, 129}, PlotPoints -> 30, 
 ColorFunction -> "Rainbow")

Enter image description here

The picture is tolerable, though not as good as the original one. You can obviously improve it by taking more PlotPoints.

But also the simplest integral

NIntegrate(g(x, y)^2, {x, -65, 139}, {y, 0, 129}) 

returns the error messages:

1: Nearest :: nearuf: The user-specified distance function dist does not give a real numerical distance when applied to the pair of points {x, y} and {-70., 0.5}.

2: Heads Nearest and List at positions 1 and 2 are expected to be the
equal.

3: "Part 1 of {} does not exist."

And other. I suspect that problem number 1 generates all other messages.

Two questions:

  1. My function seems to be very slow. Any ideas to make it faster?

  2. I can not understand what Mma does not like during the integration process.

Note: There is an alternative way to achieve my goal. It is an interpolation function to do. I have tried this, but when reviewing the display, this interpolation function appears significantly worse than the original display. Therefore I would like to check this idea first.