Hello, @columbus8myhw . I want to play a game. Up until now, you have spent your life among the dead, piecing together their final moments. By the time you read this, you will have one minute to find a finite open cover of $[0,5]$ among this cover $F$...
If p is odd prime, then any divisor of Marsene number $M_p$ = $2^p$ - 1 is of the form 2kp + 1. If q is a divisor of $M_p$ then qk = $2^p$ - 1, so we must have $2^p$ = 1 (mod q), so we must have ord(2,q) | p and by fermat little theorem $2^{q - 1}$ = 1 (mod q) hence we must also have ord(2,q) | q - 1.
I proved the following if p is an odd prime, then any divisor of Mersenne number is of form 2kp + 1.
Proof:
If q is a prime divisor of $M_p$ then qk = $2^p$ - 1 $\rightarrow$ $2^p$ $\equiv$ 1 (mod q).
gcd(2,q) = 1 since q can't be two since 2^p - 1 is always odd, so its prime divisor can't b...
@Incurrence as I remember, most derivatives act similarly, but keep in mind that for differentiation on the real line we had $\lim\limits{h\to 0}$ where we just have to check that the limits agree as $h$ approaches zero from the left and from the right, but in the case of complex differentiation, $h$ might approach zero from all sorts of directions.
I think in general, if we have a real-valued function $f$ that has an "analytic counterpart", then we can just pretend we're taking the derivative of $f$, and plug in complex values. Is that what you're getting at, @Incurrence ?
This is a theorem whose proof is not given:- I have to prove that equivalent systems have the same soloution basically they are same and homogeneous @Eoin
@LutzL I'm trying to decypher your reply to my question earlier, but I'm still having a bit of trouble figuring out exactly what to do. Aside from the central difference formulas, most of the formulas you have displayed are going over my head a bit and thus I'm struggling to apply them to my problem. Can you maybe just construct the u(t) vector and I'll see if I can pick it up from there? That's where I'm stuck at the moment.
I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have acquired the notes, the professor didn't do an example problem, which is the best way I learn a new...
if you have a linear system $A$ in $n$ variables, and say you have a solution $x_1,x_2,...,x_n$. If $B$ is another linear system that is equivalent to $A$ then it can be formed by linear combinations of equations in $A$, and then $x$ is again a solution.
If they are homogeneous, then it still holds. I'm not sure if this was what you were looking for still though.
@rememberme Maybe you wanted, say $Ax=0$ is a matrix presentation of our homogeneous system. If $B$ is equivalent to $A$ then there is a matrix $E$ of elementary transformations such that $EAx=E0$, thus $EAx=Bx=0$ and $x$ is a solution to $B$ as well.
The only other thing I have to say is perhaps it would be useful to write out an arbitrary system and an arbitrary system of linear combinations of that system. It is just a long process of bookkeeping.
A theorem of Édouard Lucas related to the Fermat numbers states that :
Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one.
Does anyone know is there some similar theorem for generalized Fermat numbers: $F_n(a)=a^{2^n}+1$ ?
I've b...
Let $\textbf{F}(x,y,z) = (xyz, x^{2} + yz)$ for $(x,y,z) \in \mathbb{R}^{3}.$ Find the first-order \hphantom{stuff}approximation of $\textbf{F}(x,y,z)$ near the point $(1,2,3).$
Well I am pretty sure you are missing some informations, and I don't really feel like wading through some other question at the moment, I am doing some things @KarimMansour
I don't know much about binary powering mod n, but how is someone suppose to give you a different algorithm when they don't know what algorithm you are using? @KarimMansour
If we answer other's question, we click the button "post uor answer" But before this we can find "community wiki" under the answer box. What is this ? If we click, what happen ?
@HKLee Community wiki is sort of giving up any "ownership" of the question, so you don't get reputations if someone votes and it people feel less squeamish about editing or adding to community wikis. Some of the main uses is for giving opinion based questions and don't feel like you should get reputation, or if a question is answered in comments and you decide to move the ideas of comments to an answer.
My theory is that now there is a long duration it takes to kick you(already shown to be longer than a day), or you now must click the leave button to leave the room
Can someone please help me what is wrong with my question. http://math.stackexchange.com/questions/1240936/solving-a-recurrence-without-initial-conditions
@Rememberme Post the problem/questions, describing what you need help with, then people can decide whether or not people want to help with whatever you problem is.
@Incurrence Yes. If the function can be written as a power series (which includes polynomials and most common smooth functions) then they are called analytic and obey all the usual rules.
I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is a typical max-min problem.
What i am trying to understand is: for different inputs, i have a dif...
@DanielFischer Hi, I don't understand the following 'fact' : why when we have a holomorphic function on a domain we have $\sup\{\vert h(z)\vert: \vert z \vert=\alpha\}=\sup\{\vert h(z)\vert: \vert z \vert\le\alpha\}$ ? I don't see the relation between this and the maximum principle.. Thanks
I don't know, I've never really been part of the community, but when I have a question, I have no problem asking someone and they don't have any problem giving a good answer.
@Incurrence: I see. I slightly remember similar behaviour taking place in high school, but definitely not at uni. There's other things to worry about at uni, no need to produce more drama, imo.
I'm interested in my grades as well, but my grade doesn't get any better if I don't answer questions from other students. At least I would think so. :P
@Huy Haha, I mean in a competitive sense. Since often grades are scaled against eachother, people will try to use diplomacy to get the maximum amount of help from people with minimum gain for others.
@DiscipleofBarney You really aren't in my class though right. You claim to be in the US, but have Australian sleep times based on your chat activity graphs
@DanielFischer (some problems with internet ) The principle is stated as if $\vert f\vert $ has a local maximum at $a\in U$ then $f$ is constant in a neighborhood of $a$.
@Gato Okay. So if $f$ is not constant, then $\lvert f\rvert$ has no local maximum in the interior. Now, $\{ z : \lvert z\rvert \leqslant \alpha\}$ is compact, so what does that imply about $\lvert f\rvert$?
@Gato Right, it attains a mximum somewhere on $\{z : \lvert z\rvert \leqslant \alpha\}$. Now there are two possibilities, either $f$ is constant, then the maximum is attained at every point, or $f$ is non-constant, and then the maximum must be attained - where?
@MikeMiller well, surely that's false? consider the inclusion $M_2 \hookrightarrow M_3$. (i was going through some old conversations and stumbled upon this message, and it felt wrong)
I proved the following if p is an odd prime, then any divisor of Mersenne number is of form 2kp + 1.
Proof:
If q is a prime divisor of $M_p$ then qk = $2^p$ - 1 $\rightarrow$ $2^p$ $\equiv$ 1 (mod q).
gcd(2,q) = 1 since q can't be two since 2^p - 1 is always odd, so its prime divisor can't b...
I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have acquired the notes, the professor didn't do an example problem, which is the best way I learn a new...
A theorem of Édouard Lucas related to the Fermat numbers states that :
Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one.
Does anyone know is there some similar theorem for generalized Fermat numbers: $F_n(a)=a^{2^n}+1$ ?
I've b...
