Recent content by xsw001

Homework Statement Find the stable/unstable manifold for the nonlinear system dx/dt=y^2-(x+1)^2; dy/dt=-x Homework Equations The Attempt at a Solution I'm trying to solve the below nonlinear system using Matlab, but got the following warning message. Any idea...

I have a general question about how to construct nonlinear ODE systems with given condition such as # of critical points with certain characteristics of the phase portrait of each critical point. I have no problem solving any type of nonlinear ODE system. But to do the reverse order, I have...

Yeah, that's true, as simple as it is. Doesn't even need any algebraic proof, just simply explain it out then.

H < CG(H) <=> H is Abelian CG(H) is the centralizer of H in G. Being a centralizer of H in G, just saying every element of H commute with every element of G. It does NOT say anything about the relationship between elements inside the H. We need to use the fact that H is a subgroup of G and...

Homework Statement G is a group, and H is a subgroup of G. (1) Show H is a subgroup of its Normalizer. Give an example to show that this is NOT necessarily true if H is NOT a subgroup. (2) Show H is a subgroup of its Centralizer iff H is Abelian Homework Equations normalizer NG(H) = {g in...

For example the Heisenberg group over Field. H(F) is a 3x3 upper right triangle where the entries on the main diagonals are all 1's. So by definition that I need to use this matrix and raise to the power where it becomes the identity matrix, then the number of that power would the...

Can anyone explain to me how to count the total # of non-invertible 2x2 matrices? I have the answer from the book, which is r^3+r^2-r provided r is a prime. But it doesn't explain how to get there, and I couldn't figure it out. I haven't been practicing linear algebra for quite a long...

Cool! Thanks tiny-tim!

Okay, for any given Sn, there are n elements in Sn {1, 2, ... m,..., n}, so we have n choices for the 1st element, then n-1 choices for the 2nd element, so on and so forth, and we have n-k+1 choices for mth element, etc. So there are total of n(n-1)(n-2)...(n-m+1)choices to form a m-cycle, to...

Homework Statement Prove that if n>=m then the # of m-cycles in Sn is given by [n(n-1)(n-2)...(n-m+1)]/m Homework Equations The order of Sn is n!. We're counting the # of ways of forming an m-cycle, then divide by the # of a particular m-cycle. The Attempt at a Solution This problem...

Homework Statement Find the order of the cyclic subgroup of D2n generated by r. Homework Equations The order of an element r is the smallest positive integer n such that r^n = 1. Here is the representation of Dihedral group D2n = <r, s|r^n=s^2=1, rs=s^-1> The elements that are in D2n...

Yes! Got it! Since t^-1=(y^-1)*(x^-1) and x^2=1 => x=x^-1, y^2=1 => y=y^-1, so t^-1=yx Then tx=xyx, and x(t^-1)=xyx, indeed they are equal! Thanks for the tips Delta! Now it seems so simple!

Let x, y be elements of order 2 in any group G. Prove that if t = xy, then t*x = x*t^(-1) Here is what I got so far. Proof: Since |x| = 2 => x^2 = 1; |y| = 2 => y^2 = 1, then (x^2)(y^2) = 1 => (xy)^2 = 1 Suppose t = xy, then t^2 = (xy)^2 = 1 WTS (want to show) t*x = x*t^(-1) This group looks...

So assume x^2=1, By definition of order of element if 2 is the smallest positive integer then order of x is 2. Otherwise 2 is not the smallest, then 1 would be the only smallest positive integer, then the order of element x would be 1. Does that look like a valid proof?

G is a group. Let x be an element of G. Prove x^2=1 if and only if the order of x is 1 or 2. How do I approach this problem? I know since G is a group, all the elements in there have the following four properties: 1) Closure: a, b in G => a*b in G 2) Associative: (a*b)*c=a*(b*c) 3)...