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I think a much easier proof would be to just note that
Note that this is just using the fact that:
And then making this swap for each n, n-1, n-2, ... except for 1.
This shows that
Are you familiar with Jordan canonical form? The proof of this is to use the JordanChevalley decomposition.
I fixed the link again, hopefully this one works. The site doesn't like directly linking to a picture because then you don't get to see their ads.
By definition i is root of -1 which satisfies the equation i^2=-1 .
The issue is that there are two numbers you can then take for the value of i, either what we currently call i or what we currently call -i. Both of these are square roots of -1.
But who's to say
is 'more' positive than ?After all, aren't they off the real-number line?
Yes, you can't algebraically order the complex numbers so that there is no sense of positive or negative. You can however lexicographically order them, in which case the natural way is that -i < i. However, if you went with -i instead, everything would work out precisely the same, modulo a few negative signs.
In the most general terms, √a can be any roots of the equation:
x^2 = a
However, standard practice is to make it the "positive" root.
You seem to be thinking that a must be positive. Also remember that in the two different equations, they are not the same 'a'.
Always go off the definition for complex log. Typically if you try to do anything else, you'll make mistakes.
So what we want to show is that
Now Euler's identity and you should be done.
Remember that once you write:
You are no longer talking about an equation. Rather, you are now talking about a set of equations.
Now try the rest and you'll see it all makes sense.
The element at the right end in C
(The largest infinity in C)in this special case, it is also Amount(C)
Please prove that there exists a "right end in C".
The gradient will tell you which way your objective function is decreasing the fastest, and is typically the way that optimization is done. In order to help further, I would need more information. Do you have a set of data points that you would like to fit? Do you have an objective function?
Also, please note that as x tends to infinity, F(x) tends to alpha, not 1.
There is rarely a formula used for such things. Not everything is easy, and in fact, most aren't. Ships all come in various shapes with different parameters, I would imagine that there is not easy way of doing this.
You're giving all the information you need in order to factor the polynomial by algebraic methods alone.
(x-1) is a factor of (x) and '(x)
This means that (x-1) is a factor of multiplicity at least 2*. Divide out by it twice using your favorite method of division and then you can easy quadratic equation the rest.
*Personally I don't remember this theorem from high school algebra, only graduate algebra when studying separable extensions. Perhaps they teach it in high school?
And there was a slight mistake on the show.
It all depends on which leaf you choose, but the standard one is to make the value pi.
The mistake that I was more annoyed at was when the show said that ≡ meant "equal by definition"
I've seen this used by a few professors. I don't see anything wrong with it. Different people take symbols to mean different things.
Sorry, Im not sure what you mean. If you want to know the names of the posters
No no, what I'm saying is that it doesn't matter who they are. If your consideration on reporting them depends on who it is, then you are doing it for the wrong motives, whether you choose to do it or not.
In other words, if I found someone doing something illegal, it wouldn't matter (at least in theory...) if they were my best friend or worst enemy. I would either always report it or never report it, based upon what action it is. Not based upon the person.
Anyone have the faintest idea why this thread has over 35,000 views?
Why is your acceleration
and not just
The description of the person only serves as a distraction. If who the person is matters for your judgment of what to do, then no matter what you do you have the wrong motives.
Even when not knowing who the person is at all, the question can still be an interesting one.
It looks like they just "rounded". Companies don't advertise exact figures, typically they make it a more appealing figure by putting a zero (or more) in some digits.
The pairing is not complete. And this is why the proof is false. (this false proof fooled numerous folks to accept one set and its subset can have the same amount of elements just because they cannot prove it wrong)
Name an element from set B or C that is not paired.
Still shouldn't a game be in keeping with scientific and economic possibility.
Why restrict yourself to the real world?
Also the rainforests contribute about a quarter of the earths oxygen.
Where is that number from?
Soroban, that sounds frighteningly familiar to a common creationist argument. See this video.
I corrected a typo in my last line, the last term was x_j and it is now x_n.
This is not a definition, it's just using the standard definition of matrix multiplication on the vector (0, ..., 1, ..., 0).
So it could have been row(j) elements, but they are equivalent?
No, whenever doing matrix multiplication (i.e. applying a linear transformation), vectors are always column vectors.
The page in question is here.
By "x_j" what the author means is the vector (0, ..., 1, ..., 0) where the 1 is in the jth component. This is of course relative to the basis, so the real vector (not written just as components) is:
Hopefully that makes sense. Now multiplying this element by the matrix should reveal the vector:
But since we write this in terms of the basis {x_i}, this is really:
As the author claims.
What book?
Since you seem to have no objection to my statement that dominance has nothing to do with being able to kill any species (if you do, please let me know), then let us move on from this, and get back to talking about dominance.
As I posted a little while back, dominant means the most in the control. So there is always at least one (though possibly multiple) dominant species. If you (you, meaning everyone reading this) think that humans are not the dominant species, then I would like to hear suggestions as to which species are more dominant, and your reasons why.
I haven't managed to stump it yet, but maybe I'm not giving it exotic enough symbols.
It wasn't recognizing sharp (i.e. tic-tac-toe board) until I trained it a few times. That thing learns pretty darn quick!