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First prove that no element can be of order 9. Then for any element of order 3, take a look at the set:
<Z(G), x>
How many elements in that subgroup? How many different ways are there to do this (distinctly!)? And finally, could there be any subgroup of order 9 that does not take this form?
Think of a circle centered at 0 of radius |r(t0)|, what is the radius of curvature?
Now think of your path, r(t), as it approaches the point r(t0). Where does it approach from (in the circle? Outside of the circle? On the boundary?), and given this, what can you conclude about it's radius of curvature?
Hint: The probability of getting that last card to be of the correct suit was (13-4)/(52-4)
Let G be a group of order 6. An easy way is to use Sylow's theorem which proves that G is a semi-direct product of Z2 and Z3, which can give rise to at most 2 groups. And here they are: S3 and Z6.
But this uses a bit of machinery. If you know group actions, then you can let G and on H, a nonnormal subgroup of G of order 2. Prove that this action is faithful, and then of course it goes into S3, and you're done.
It is also possible to prove that G = {1, x, x^2, y, z, w} where x is of order 3, y, z, and w all have order 2. Then using the fact that G is a subgroup of S4, you should be able to prove that y, z, and w are all transpositions. Given this, it should only take a few calculations to show that x, y, z, and w are permutations of only 3 letters (numbers).
There are only two groups of order 6 up to isomorphism: S3 and Z6 (note D3 is isomorphic to S3).
Robert?
Just because a term is unbounded (for each M there exists a choice of x, y, and z such that...) does not mean that it holds for every choice of x, y, and z.
If you understand the proof of De Morgan's rule for just two variables, then use induction to get the general statement.
There are many different aspects to the quaternions, as they are used is many major branches of mathematics. You would need to be a bit more specific with what you want to do.
As for visualizing the complex plane, what exactly do you mean? I have a feeling it might be (Mobius, Linear-fractional) transformations of the complex plane?
While longer in the short run, this method will give you faster overall convergence.
Use the taylor series not for ln(2), but for e^x. It converges much faster. Now that we can calculate e^x, we can use Newton's method:
Starting out with x_0 = 1, I get 0.6932336 where e^x is calculated using the first 6 terms of the taylor series and 3 newton steps.
For your 2nd question, if that's the order they have it, then yes it is a mistake. Note that you must integrate over theta first.
The theorem is known as Fubini's theorem. You may ignore the hypotheses, they are all satisfied by your integral.
Jane, you need to calm down. I think it's clear that mathsyperson intended to convey that there are an infinite amount of ways to write these roots, especially considering the fact that e^(i*t*pi) is periodic for real variable t.
You can't get mad if someone makes a mistake. It hurts the forum when you do, and it encourages others not to post. But I've said this a million times before and I know you have no intention of listening this time.
A much easier way to solve these inequalities is to find all the places where they are either:
(i) Equal
OR
(ii) discontinuous
This will divide up the real line into intervals, and then you just test each one.
From what I can tell, there is no phenomena called "super solar winds".
just like when you flush the bathtub, a sundot can form on top of another.
I assume you mean sunspot, but how is that at all like letting water drain from a bathtub?
If it has opposite direction I believe the void of wind is filled with the chaos of the sun and release a solar wind.
Ok, so now sunspots have a direction. I assume your are talking about the "wind up" phenomenon? And somehow the solar wind is affected by sunspots?
moreover I believe that when a solar spot is created within another, with opposite direction, that a solarwind is formed.
There is no evidence to suggest that solar wind is correlated with sunspots, and because of all the measurements of the sun we do, if it was, we would have found it by now.
so basically there's a huge hole where it is, vector wise.
Please define "vector", as I am not familiar with your usage of that word.
You wouldn't see a hole form in a solid diamond here on earth, in the same way if you empty your bathtub, the hole is the most likely suspect. that's the analogy
I think your argument is, "It happened their once, so that's the most likely spot for it to happen, and therefore it will happen their again." I won't pretend to know the dynamics of sunspot creation, but you really shouldn't either. It's a complex system with many variables which is not yet fully understood. Your attempt at an analogy ain't alright. All alliteration aside, it falls way short.
the bathtub flush through the hole...
I'm going to guess that this is a metaphorical bathtub. A metaphorical bathtub that flushes. Did you mean a toilet?
...and the solarwind eminates when...
By "emanates", did you mean to suggest that this is what causes the solar wind?
That happens if 2 solarwinds are created one within the other without having the same direction.
Did you mean that two solar winds come together to form one?
Your ideas here are incomprehensible. I'm trying, and I mean really trying, but it's impossible to figure out what it is you're trying to say.
Are you trying to predict the year that something will occur based on knowing it's percentage of chance of occurring?
One thinks there is another black hole.
One would be correct in one's assessment.
It existed when the particles were first created, it just wasn't measured.
This is one of the hardest things to understand (or perhaps, simply become accustomed to) in quantum physics. The thing is that it didn't exist when the particles were created. There are several states (spins) a particle can have, and we can express the probability that a certain particle is in a certain state, called the wave-function. When the state is actually measured, we know with certainty which state the particle is in, and the wave-function is said to "collapse".
All this is perfectly fine, except for one thing. Scientists currently think (and experiments have given them good reason to) that the wave-function is not just based on our incomplete knowledge, but the "probability aspect" is actually a physical phenomenon.
But like i said, information cannot be transferred, because you cannot control what the spin on your particle is when it is observed, all you know is that the other particle must have the opposite.
You're saying it's not information because you can't change it?
Works as soon as I put the links on two different lines.
The error duplicates here. Copy my text below, and see if you can move the 2nd link up into the first paragraph.
Any (elementary) number theory, discrete math, or intro to proofs book will contain material on modular arithmetic. If you want to just learn modular arithmetic, I would go with sources on the web. Rutguers has an introduction, but there are