Recent content by hypermonkey2

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    Length of parallel curve

    wicked! I get it now, thanks everyone! cheers
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    Length of parallel curve

    Hi, I found this problem in Do Carmos "differential geometry of curves and surfaces". it asks to show that the length of a parallel curve B to A given by: B=A-rn where r is a positive constant, and n is the normal vector, and A is a closed convex plane curve, positively oriented. is...
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    Imaginary parts of roots of unity

    Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!
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    Splitting Fields

    Very true! However, what does 2^(1/4) mean exactly in this case? i dont think it can be a real number since i dont believe there is an extension from F_5 to R... And since there is no element x in F_5 such that x^4=2... perhaps i am confused? Thanks for you reply!
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    Splitting Fields

    I was thinking about this, finding the splitting field of x^4-2 in Q[x] over Q is standard enough... but would much be different is i wanted the splitting field over F_5? (field with 5 elements) would it just be F_5(2^(1/4), i) analogously to the Q case? or do any of the arguments break down...
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    Similarity classes for matrices

    Homework Statement How would we find number of similarity classes for a nxn matrix over the field Fp (cyclic of order p) for n=2,3,4? Homework Equations A and B are similar iff they have the same monic invariant factors The Attempt at a Solution I would say 4 classes for n=2, since...
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    Torsion-free modules

    ah so i see why it needs to be torsion free. (M lies in R, so if there is r such that rm=0, R cant be an integral domain...) but what about the rank? Why must it be 1? I am also surprised that it is not free..
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    Torsion-free modules

    Hi all, I came across this problem in a book and I can`t seem to crack it. It says that if we have an integral domain R and M is any non-principal ideal of R, then M is torsion-free of rank 1 and is NOT a free R-module. Why is this true? cheers
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    Connectedness of coordinates with one rational point

    Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped. Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected. My gut says that it should be path-connected too (thus connected), but...
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    Sequence of closed sets

    is it even reasonable to say that any closed set in this space will have a closed sphere containing it? And if so, can we simply define a set this way and take the infimum along the radii?
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    Sequence of closed sets

    of course! yes i apologize i mean in a complete metric space where the diameter of each set in the sequence is bounded. Thanks for pointing that out :D so how would i go about finding these spheres?
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    Sequence of closed sets

    thanks for the reply! it is true what you write, but i am having trouble making the connections... What does this imply? cheers!
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    Sequence of closed sets

    essentially, my question can be boiled down to: for any closed set, can we find a smallest closed sphere containing it? what about a smallest closed sphere contained IN it?
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    Sequence of closed sets

    I was reading about the Nested sphere theorem and a thought occurred. if you have a sequence of decreasing closed sets whose diameter goes to zero in the limit, we can show that the intersection of all these sets is a single point. my idea was to show this using nested sphere theorem if we...
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    Origin of the term seperable

    Origin of the term "seperable" I was just curious as to why out of all properties of metric spaces (ie compactness, closure, etc), i dont know how the term seperable makes sense intuitively.... is there an origin to this term? Just curious. cheers
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