Recent content by Ad123q

I'm reading a paper and have came across the term 'Cn-close' in the sense of a curve being C1-close to a circle for example, but can't find a definition of this term anywhere, and would be grateful if anyone could help.

I'm not sure if my reasoning below is correct or not. If a=e\stackrel{\underline{2πi}}{5}, then Q(a) = {r + sa + ta2 + ua3 +va4 : r,s,t,u,v \in Q} . [Is this correct?] Then [Q(a):Q] = 5 as {1, a, a2, a3, a4} form a basis for Q(a) as a vector space over Q. However I am not sure if my...

I have in my notes that (2X,5) is an ideal of Z[X], but I can't see why this can be so. For example 5+2X is in (2X,5) and 7+X is in Z[X] but then (5+2X)(7+X) = = 35+5X+14X+2X^2 = 2X^2+19X+35. 19 is not divisible by 2 and so this element is not in (2X,5), contradicting the...

Hi, Was wondering if anyone could give me a hand. I need to prove that the Cayley Transform operator given by U=(A-i)(A+i)^-1 is UNITARY, ie that UU*=U*U=I where U* is the adjoint of U (I am given also that A=A* in the set of bounded operators over a Hilbert space H). My solution so...

I have in my algebraic topology notes, as a step in the proof of another theorem, that the product of a finite simplicial complex X with a single point (a 0-simplex) is isomorphic to the finite simplicial complex X, but I can't see why this is so. i.e. Xx{point} isomorphic to X Thanks

Hi, This has came up in a proof I'm going through, and need some guidance. The proposition is that if R is a principal ideal domain, then every submodule of a free module is finitely generated. The proof starts let F isomorphic to R^n be free, with basis {e1, ... , en}. Let P be a...

Thanks, though I'm still not 100% sure about how to get this. Using ratio test I obtain mod[ (z-e)^(n+1)! / (z-e)^n! ] = mod[ (z-e)^n!(n+1-1) ] = mod[ (z-e)^n.n! ] Then I'm not sure where to go from here.

Hi, am a bit stuck with this. Find the radius of convergence of the complex series (Sigma n=1 to infinity) (z - e)^n! I know that the answer is R=1 but I'm not sure how to get there. It's the factorial as a power which I'm not sure about, have seen this in some other problems too. I...

Anyone?

Sorry, just realized that I didn't in fact highlight what I don't understand! Basically I don't understand the whole reason why we can go from v_n+1 = Av_n to v_n = A^nv_0 (ie the process to do this); is it induction of some kind? This type of problem must be fairly easy but I can't see it...

Hi Was wondering if anyone could help me with this topic, which relates to the solution of simultaneous linear difference equations and diagonizable matrices. Sorry for the lack of latex, but as such I will denote an exponent by "^" and a subscript by "_". Say we have a system of...

Hi There are two things that are confusing me a bit, and was wondering if anyone could explain them. Firstly, if we let V1,V2,...,Vn be vectors in some field F^n and let P = {V1,V2,...,Vn}, then the following are equivalent: (i) {V1,V2,...,Vn} is a basis for F^n ; (ii) P is...

Hi, I'm slightly confused about one aspect of the conditions for applying L'Hopital's rule. N.b. apologies in advance for the lack of LaTeX. L'Hopitals rule: Let f,g:(a,b) → R be differentiable and let c ε (a,b) be such that f(c)=g(c)=0 and g'(x)≠0 for x≠c. Then lim(x→ c)[f(x)/g(x)]...

Thanks, so if I was to write x1, x2, x3, x4 under the matrix in question, would the solution space for the general solution just be x = {x1 x2 x3 x4} = {x1 0 0 0} ?

I was just wondering, if I had a matrix in reduced-row echelon form, say, 1 0 0 2 0 1 3 1 0 0 0 0 then I could write the general solution as a1= -2a4 , a2= -3a3-a4, with a3 and a4 defined in terms of these. (I obtained this solution by putting a1, a2, a3 and a4 under the first...