What is D?(Original post by Indeterminate)
Problem 372***
Prove the following:
If
and if f is holomorophic and
then

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 12102013 16:53

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 12102013 17:55
(Original post by Hodor)
What is D? 
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 12102013 18:17
(Original post by james22)
I assume it's any domain in the complex numbers on which f can de differentiable. 
FireGarden
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 12102013 18:31
(Original post by Hodor)
Then I don't think it's true.
You'd know this was constant, since the tangent to the curve of f is always parallel to the x axis  which only happens when f is constant. The above proposition says the same is true for complex functions; and by the definition of the complex derivative, it should appeal to intuition that it is 'obviously true', as any straight line in the complex plane behaves like the real line in terms of differentiation (i.e. is not really a special line for differentiability; complex derivatives must exist and be equal in all possible paths of approach). 
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 12102013 18:48
(Original post by FireGarden)
..Really? If you have a real function which was differentiable everywhere, and =0 everywhere, wouldn't you believe that was constant?
You'd know this was constant, since the tangent to the curve of f is always parallel to the x axis  which only happens when f is constant. The above proposition says the same is true for complex functions; and by the definition of the complex derivative, it should appeal to intuition that it is 'obviously true', as any straight line in the complex plane behaves like the real line in terms of differentiation (i.e. is not really a special line for differentiability; complex derivatives must exist and be equal in all possible paths of approach). 
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 12102013 19:17
(Original post by Hodor)
In the real case, let A = (1,0) u (1,2). Then I can think of a nonconstant function f: A > R with f' = 0... 
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 12102013 19:55

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 12102013 20:09

FireGarden
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 12102013 20:28
(Original post by Hodor)
In the real case, let A = (1,0) u (1,2). Then I can think of a nonconstant function f: A > R with f' = 0...
Anyway, some more 'different' maths. Group theory!
Problem 373***
Let be a group, which acts on the set .
Suppose have the same orbit. Prove that their stabilisers are conjugates (i.e.
Problem 374***
How many ways can you colour the vertices of a square using at most three colours, which are distinct under the action of ? (i.e. colourings which cannot be obtained from another by rotations or reflections) 
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 12102013 22:42
(Original post by FireGarden)
Anyway, some more 'different' maths. Group theory!
Problem 373***
Let be a group, which acts on the set .
Suppose have the same orbit. Prove that their stabilisers are conjugates (i.e.
Solution 373
Simple.
From the definition of an orbit, if lie in the same orbit, then, there exists a that acts on such that, .
Now, suppose,
.
Therefore, we have, .
However, repeating the steps above using , we have,
.
And so, by symmetry, we can assert that, .Last edited by MW24595; 13102013 at 00:52. 
Indeterminate
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 13102013 00:51
(Original post by Hodor)
X(Original post by james22)
X(Original post by Mladenov)
X
Oops, as I had posted it from memory (came across it a long time ago), I made some mistakes in stating it. I'll consult the source and repost it 
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 14102013 21:32
(Original post by FireGarden)
Problem 374***
How many ways can you colour the vertices of a square using at most three colours, which are distinct under the action of ? (i.e. colourings which cannot be obtained from another by rotations or reflections)
Let be the set of colorings of the vertices.
Firstly, observe that two colorings are essentially distinct (i.e. distinct under the action of the dihedral 8 group) if they belong to different orbits of in .
Hence, we shall be done if we manage to find the number of orbits of in the set of colorings.
The following result is easily proved (counting in two ways, Lang's Algebra, ect.)
Proposition:
Let be a finite group acting on a finite set , and is the number of elements such that . Then the number of orbits of in is .
Since the representative does not matter, we have and hence the number of orbits of in is .
Fix . Notice that there is a bijection , call this . Also, the number of such that is the order of .
We count the number of solutions to the equation in two ways.
One the one hand, there are elements such that . On the other hand, each is counted times by the sum since there are 's in such that . Thus ; call this .
Hence, by and , we obtain the result.
We can interpret the elements of as products of cycles.
Second Proposition (too general):
Let be a product of cycles (counting cycles of length ). Then the number of colorings fixed by is , where is the number of available colors.
Although, we do not need it, I write it in case someone is interested.
For our problem, it is sufficient to note that the vertices in a given cycle have the same color.
Thus, if colors are available, there are essentially distinct colorings, since the elements are , , , , , , , .
As a side note, we have a problem with the numeration of the problems. It starts with Indeterminate's problem 372.
Basic commutative algebra.
Problem 377***
Prove that if is an matrix over a local ring, with coefficients in the maximal ideal , then is invertible.
Problem 378***
If is a surjective endomorphism of a Noetherian ring , then is an isomorphism.Last edited by Mladenov; 14102013 at 23:11. 
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 16102013 10:59
Last edited by Zishi; 16102013 at 11:01. 
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 16102013 12:50
(Original post by Zishi)
Sorry, probably my maths skills have gotten rusty  I can't understand the part after "hence...". I is , but how does that become ? 
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 16102013 14:43
(Original post by james22)
You have 2 experssions for I, one has sine terms on the top and the other cas cos terms. By adding them you get (sin^17+cos^17)/(sin^17+cos^17)=1 so 2I=1 
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 28102013 20:22
Problem 379 **
How many subsets of are there such that the sum of the elements exceeds ?
Is this possible to generalise? 
The nameless one
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 29102013 14:44
(Original post by henpen)
Problem 379 **
How many subsets of are there such that the sum of the elements exceeds ?
Is this possible to generalise? 
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 29102013 23:57
(Original post by henpen)
Problem 379 **
How many subsets of are there such that the sum of the elements exceeds ?
Is this possible to generalise?

Kvothe the Arcane
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 30102013 00:06
(Original post by Mladenov)
It can be generalized. We have to consider two cases  .
What do you think? Using and , we are done.
Last edited by Kvothe the arcane; 30102013 at 00:30.
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