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...
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...
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)...
Homework Statement
I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces.
Homework Equations
If a function f is continuous in metric space (X, d), then...
Yeah, that's what I was thinking originally instead of use a particular example, just generalize it by differentiating k times. I got pretty close the same answer,
p(a)=p’(a)=p’’(a)=…=[p^(k-1)](a)=0, but (p^k)(a)=k!r(a) not equal to 0 since r(a) not equal to 0 and k is natural number.
Should...
a is a root of order k of the polynomial p provided that k is a natural number such that p(x)=[(x-a)^k]r(x), r is a polynomial and r(a) not equal to 0.
Prove a is a root of order k of the polynomial p iff p(a)=P'(a)=...=[p^(k-1)](a)=0 and [p^(k)](a) not equal to 0.
Note:
[p^(k-1)](a) :=...
L(f,P) is the Lower Darboux Sum; L(f,P)=sum mi(Xi-Xi-1)
U(f,P) is the Upper Darboux Sum; U(f,P)=sum Mi(Xi-Xi-1)
f: is the function
P: is the partition of the domain [a,b]; P={Xo, X1, ... , Xn}
mi: is the greatest lower bound [Inf(f)]
Mi: is the least upper bound [Sup(f)]
delta Xi=Xi-Xi-1: each...
Suppose f, g:[a,b]->R are bounded & g(x)<=f(x) for all x in [a,b]
for P a partition of [a,b], show that L(g,P)<=L(f,P)
I don't know whether I should show by cases since I don't know the monotonicity of the both functions f and g. It seems like that the graphs of both functions have to behave...
Suppose f:[a, b]-> R is bounded function
f(x)=0 for each rational number x in [a, b]
Prove Lower Integral <= 0 <= Upper Integral
Proof:
f(x) = 0 when x is rational
both L(f, p) = U(f, P) = 0
and L(f, p) <= Lower Integral <= Upper Integral <= U(f, p)
This function seems like discontinous even...
1. Homework Statement
Prove that a point xo in Domain is either an isolated point or a limit point of D.
2. Homework Equations
xo in D is an isolated point provided that there is an r>0 such that the only point of domain in the interval (xo-r, xo+r) is xo itself.
3. The Attempt at a...
Yes, you're right.
Well, limit point of a set D\{xo} is a number Xo such that every deleted delta neighborhood of xo contains members of the set. For any delta>0, we can always find a member of the set which is not equal to xo, such that |x-xo|<delta.
D={set of real numbers consisting of single numbers}
Show set D has no limit points, and show the set of Natural numbers has no limits points.
I know it's a very simple question. I don’t know my way of approaching this is appropriate or not. Let me know. Thanks.
A finite set of real...
Here is the sketch of the proof.
Assume by contradiction that a<c<b, and case 1) f(a)<f(b)<f(c) or case 2) f(c)<f(a)<f(b)
Since the function is one-to-one, therefore the graph of the continuous function can't oscillate, so it is strictly monotone, either strictly increasing or decresing.
case 1)...
Okay, I'm still not completely getting the right approach to complete this proof. "fzero", you were saying that I need to use the fact of f is 1-1, along with f is continuous on [a, b] that only implies the function is NOT a constant function, so it is strictly monotone, isn't it?
The IVT...