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...
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!
Very true! However, what does 2^(1/4) mean exactly in this case? i don't think it can be a real number since i don't 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!
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...
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...
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 can't be an integral domain...)
but what about the rank? Why must it be 1? I am also surprised that it is not free..
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
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...
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?
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?
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?
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...
Origin of the term "seperable"
I was just curious as to why out of all properties of metric spaces (ie compactness, closure, etc), i don't know how the term seperable makes sense intuitively... is there an origin to this term?
Just curious.
cheers