What does the ‘ (prime) mean in this context?

I’m trying to check a simple calculation regarding the derivative of a matrix exponential. By hand I get $ $ \frac{d}{d\theta_1}e^{\theta_1 \mathbf{W_1}+\theta_2\mathbf{W_2}} = \mathbf{W_1}e^{\theta_1 \mathbf{W_1}+\theta_2\mathbf{W_2}}. $ $ Checking this with Mathematica, I have

Assuming [{\[Theta]1 \[Element] Reals, {W1, W2} \[Element]     Matrices[{d, d}, Reals, Symmetric]},   D[MatrixExp[ (\[Theta]1 W1 + \[Theta]2 W2)], \[Theta]1]] 

which returns

(* W1 Derivative[1][MatrixExp][W1 \[Theta]1 + W2 \[Theta]2] *) 

Why is there a derivatie here? And what’s the [1]?

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An efficient way of calculating ๐œ™(๐œ™(p*q)) where p and q are prime

Let p and q be prime numbers and ๐œ™ Euler’s totient function. Is there an efficient way of computing ๐œ™(๐œ™(p*q)) = ๐œ™(๐œ™((p-1)(q-1)), that is not simply based on factoring (p-1) and (q-1)?

Obviously, if p and q do not equal two, (p-1) and (q-1) are even and consequently their prime factorization is entirely different from the prime factorization of p and q. Therefore I assume that no such shortcut exists.

Do I overlook something?

How are prime implicates of HORN-Formulas defined?

I’m confused about the definition of prime implicates in Horn formulas.

For example in the paper of Kira 2012 on page 109 it is stated: enter image description here

Now in the paper of Boros 2010 on page 82 the following definition is used: enter image description here

My goal is to decide whether a Horn formula is prime or not in polynomial time. For that I want to assume the definition used in Kira 2012.

How can I prove that the two definitions above for prime implications of Horn formulas are equivalent?

Prime sphere requirements to use Tass in Mage 20th Anniversary Edition

I’m confused about what level of Prime is required to use Tass in the Mage 20th Anniversary Edition rules.

In the main rulebook the following statements are made:

M20 Rulebook:

  • p 332: “To absorb Quintessence energy from the materialized Quintessence of Tass requires Prime 3.”

  • p 520: Prime 1 sphere description says: “A beginning study of Prime allows the mage to perceive and channel Quintessence from Nodes, Tass, …”

  • p 520: Prime 3 sphere description says: “… the Prime-skilled mage can draw both free and raw Quintessence from Nodes, Junctures (special times) and Tass …”

How do you DO That Rulebook says:

  • pp. 43-45: “Any mage with an Avatar Background cab absorb that Background’s worth of Quintessence from a Node or Tass …”

First off – are there ways to use Tass other than absorbing the Quintessence into your pattern? Can you spend Tass to gain Quintessence effects like lowering the difficulty of a roll without absorbing it into your pattern?

And with regard to what levels of Prime are required to do these things, I can see support for any of these positions:

  1. You don’t need any Prime at all to absorb Quintessence from Tass into your pattern, as “Any mage with an Avatar Background” can do it (How do you DO That pp. 43-45)
  2. You need Prime 1, as the description for that sphere says Prime 1 allows you to channel Quintessence from Tass. (M20 p. 520)
  3. You need Prime 3, as that is stated on M20 p. 332 and p. 520 explicitly

I have reviewed other answers at: How does the Prime Sphere affect gaining and storing Quintessence in M20? but it left me more confused than enlightented.

From a gameplay perspective, requiring Prime 3 to use Tass seems to defeat much of the purpose of Tass – as it can’t be readily exchanged between mages as portable Quintessence unless they have Prime 3.

how to send prime and generator of diffie hell-men to client over network node js?

I am using crypto module of node js for exchanging key using diffie-hellman algorithm.

server.js

const crypto = require("crypto");  const alice = crypto.createDiffieHellman(512);  const aliceKey = alice.generateKeys(); 

client.js

const bob = crypto.createDiffieHellman(alice.getPrime(), alice.getGenerator());  const bobKey = bob.generateKeys();  const aliceSecret = alice.computeSecret(bobKey);  const bobSecret = bob.computeSecret(aliceKey); 

The above example is taken from node.js documentation as shown the client uses servers prime number for generating the prime number.

my question is how should I securely send the prime number and the other parameter to client over internet? are there any other alternatives?

and another question is that I am generating keys using generate keys function but I have already generated private-key.pem and public-cert.pem file. can I use those if yes then how?, if no then what is difference between those keys?

Derivative with prime for vector-valued function

Consider first the following vector-valued function of a real variable:

   s[t_] := {Sin[t], Cos[t]} 

Then this works as expected:

   s'[t] (* {Cos[t], -Sin[t]} *) 

Why does the following use of prime to take derivative not also work?

   soln[t] := {x[t], y[t]} /.    First@ DSolve[{Derivative[1][x][t] == y[t],       Derivative[1][y][t] == -x[t], x[0] == 0, y[0] == 1}, {x[t],       y[t]}, t]     soln[t] (* {Sin[t], Cos[t]} *)     soln'[t] (* soln'[t] *) 

Note that the following does work:

   D[soln[t], t] (* {Cos[t], -Sin[t]} *)