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.


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


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]} *) 

Show/prove that $UPrime = \{ |^n : n ∈ N \space is \space prime\}$ is in P

I have this question to solve. According to my understanding, it basically requires a turing machine that outputs lines on the tape, with the number of the lines being any prime number.

My idea is to take the AKS test’s conclusion and making a case that since, calculating primes is a problem that can be solved in polynomial time complexity as already proven by AKS, hence this problem is also in P.

Is this the right way? what would be a more formal/mathematical way of expressing this if it is?

Pumping Lemma with Prime Number

$ \text {Could someone please help me with this proof: }$ $ L:=\left\{a^{n} d^{m} b^{k} | n, m, k \in \mathbb{N} \wedge m \text { is a prime number}\right\}$

$ \text {Maybe we can say, that } w=a^{n}d^{m_s}b^{k} \text { and } d^{m_s} \text{ is a prime number with } |w| \ge n. \text{And we choose xyz arbitrarily with } w= xyz \text { } \wedge |y| \ge1 \wedge |xy| \le n\text {. And now I don’t know how to go on. Maybe someone can help me, please.}$

Can I use a salad fork to get into the Prime Material Plane?

In several editions of Dungeons and Dragons, the Plane Shift spell requires a forked metal rod, the design and material of which varies depending on the destination plane. Is there any canonical list of which rod designs are required for different planes—in particular, the Prime Material?

(I’m using 3.5, but I’ll accept sources from other editions.)

How can I kill/separate a Deity from the Prime Material Plane?

I need my BBEG to kill/separate a god from the material plane/everything. The PCs are helping NPCs save their god, and while I think they could probably do it, I do wanna know what to do if they don’t or if they take too long (seeing as the NPCs lose their divine shit).

The deity isn’t on the prime material plane, but she has a source of power/influence that the BBEG is trying to destroy/whatever. Is there an established way to either trap the god somewhere so she can no longer contact/influence the material plane, or just straight up kill her? (Alt, if there aren’t any established ways, any tips on how to set it up?)

(This is my/our first time playing D&D, so I apologize if I’m not making much sense. Thanks!)