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 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!
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 cant 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 dont know how the term seperable makes sense intuitively.... is there an origin to this term?
Just curious.
cheers
Homework Statement
find x if sin(x)=1/5
Homework Equations
x=arcsin(1/5)= ??something in closed form maybe??
The Attempt at a Solution
This actually isnt my homework. A student im tutoring in differential calculus came to me with this problem and im convinced it is a typo. Just for...
They inspect the product. Essentially, i think that IF we assume that they perform some sort of inspection, that IF one inspector deems it defective, that it is MORE likely that the product actually IS defective, so it is MORE probable that the second inspector deems it defective too.
In that...
Heres something i came across in a book but theres no solution...
take two inspectors in a factory (they cant talk to each other), and they inspect a series of products and they deem them defective or not. So the results would be (D,N), (N,N) ..... where each coordinate is each inpectors...
Homework Statement
Show that d and p are equivalent metrics on X where p=d(x,y)/(1+d(x,y))
Homework Equations
ive proved already that p is indeed a metric too (if d is a metric).
The Attempt at a Solution
I believe im supposed to use the Lipschitz condition where there exits...
hey guys, i came across this inequality in analysis and am not sure how to prove it. Any ideas? It's not homework, im just curious..
Let a1, a2, . . . , an be strictly positive real numbers. Show that
a1 + a2 + · · · + an−1 + an <= ((a1)^2)/a2 + ((a2)^2)/a3) +...+ ((an)^2)/a1
cheers
in either case, the jacobian gives me an expression that i can only integrate with mathematica using error functions or elliptical integrals... im starting to think that this is impossible!
So i just let x=rcos(o)sin(phi) etc etc? i wouldnt need to try something like x^2=rcos(o)sin(phi)?
Because i dont see how to find limits for r in the first case...
Homework Statement
What would be the most efficient way to find the volume of the solid x^4+y^4+z^4=1?
Homework Equations
The Attempt at a Solution
Cylindrical and spherical coordinates end up messy with integrals that cannot be computed by hand. Im at a loss to find something...
Thats sweet! Just what i needed. Yup, the second is true all right. The derivative method works fine by taking the first three derivatives, but its a little messy and only works on (1, infinity). I managed to find something on (0,1] but its also messy. Any elegant ideas?
hmm no that wont do. for the first, is it enough to say that since at x=0 both sides are equal and the derivative of one is always greater than the other? A similar reasoning might work for x>1 by taking the first 3 derivatives...