(1) Show that any C1 vector Field on S2 (the torus) possesses at least one singularity.
(2)Show that any isolated periodic orbit T of a C1
planar vector field X is a limit cycle.
Any help/suggestions are appreciated.
Thanks for the response Dick.
If f_{n}=I_{(\frac{n-1}{n},1)}, then \left\|\underline{lim} I_{(\frac{n-1}{n},1)}\right\|_{\infty}= 0 < \underline{lim}\left\|I_{(\frac{n-1}{n},1)}\right\|_{\infty} =1
Please correct me if I am wrong.
correction
Problem: Show an example of a sequence of measurable positive functions on (0,1) so that
\left\|\underline{lim} f_{n}\right\|_{\infty} < \underline{lim}\left\|f_{n}\right\|_{\infty} for n\rightarrow\infty
My work: I think its just the indicator function I_{[n,n+1]}
Since...
Problem: Show an example of a sequence of measurable positive functions on (0,1) so that
\left\|\underline{lim} f_{n}\right\| < \underline{lim}\left\|f_{n}\right\| for n\rightarrow\infty
My work: I think its just the indicator function I_{[n,n+1]}
Since \left\|\underline{lim}...
triangle inequality for measures.
Theorem: \mu ( \left\{ \Sigma f_{n} > \Sigma \epsilon_{n}\right\} ) \leq \Sigma \mu(\left\{ f_{n} > \epsilon_{n} \right\})
Proof: If for all n we have f_{n} \leq \epsilon_{n} then
\Sigma f_{n}(x) \leq \Sigma \epsilon_{n} and so
\bigcap...
I agree it is not a simple fact. I did not thoroughly go through that result, I am more interested in seeing the idea(punchline) in the proof you kindly posted.
Problem: f_{n}\rightarrow f in measure, \mu(\left\{f_{n}>h\right\})\leq A
Prove that \mu(\left\{f>h\right\})\leq A.
My Work:
Suppose not, then \mu(\left\{f>h\right\}) > A.
From the triangle inequality for measures we get
\mu(\left\{f>h\right\}) =...
Homework Statement
If T is a nilpotent transformation from V -> V, V - finite dimensional vector space.
show that a_{0}+a_{1}T+\cdots+a_{k}T^{k} is invertible. a_{0} nonzero.
Im having trouble finding the inverse, I know for 1+T+\cdots+T^{m-1}
the inverse is (1-T),where T^{m}=0. I also...
Homework Statement
If a and b are algebraic over F of degree m and n, both relatively prime, then
F(a,b)=mn, (i.e. [F(a,b):F]=mn)
any comments are helpfull.
Homework Statement
Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
and g has degree n.
Its clear that F[x]/( g(x) ) is a vector space and that
B= (1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ),
but I am having trouble showing that B is linearly independent...
Homework Statement
Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
and g has degree n.
Its clear that F[x]/( g(x) ) is a vector space and that
B= (1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ),
but I am having trouble showing that B is linearly independent
[SOLVED] complete measure space
Homework Statement
Assume that (\Omega,\Sigma,\mu) is a complete
measure space, let \mu_{e} be the outer measure defined by \mu
. Prove that if \mu_{e}(S)=0 \Rightarrow S\in\Sigma .
I know that \mu_{e} = \mu when restricted on \Sigma
and that if...
Homework Statement
Let D be the collection of all finite subsets ( including the empty set) of [0,1].
Prove that D is a semi-ring. What is \sigma(D) ? Define on D: \mu (A)=#A . Prove that \mu is a premeasure and identify \mu_{e} and
\Sigma_{mu_{e}} . Is ([0,1],\sigma (D), \mu_{e}) complete...