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...
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...
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...
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...
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...
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) :=...
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...
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...
f:R->R is odd, if f(-x)=-f(x) for all x
Show if f:R->R is odd and the restriction of this function to the interval [0, infinity) is strictly increasing
Then f:R->R itself is strictly increasing
I’m very confused about what the question is exactly asking for. From my understanding of the...
Let f:[a,b]->R be continuous and one-to-one such that f(a)<f(b).
Let a<c<b. Prove that f(a)<f(c)<f(b)
My first instinct is to apply intermediate value theorem. Let me know whether my proof makes sense or not.
Proof:
Since f:[a,b]->R is continuous and one-to-one.
Therefore f is strictly...
I have two general questions that I'm NOT sure if it's absolutely accurate statement or NOT.
1) Odd function is always strictly monotone, either strictly increasing or strictly decreasing right? If there any counterexample to disprove my assumption?
2) One-to-one function is always...
1) Find a continuous function f: (0,1)->R with f[(0,1)]=R
I couldn't think of any function except tangent, but its domain is NOT (0,1) though? Any suggestions?
2) Find a continous function f: (0,1)->R with f[(0,1)]=[0,1]
I couldn't think of any function that I know. Any suggestions?
3) Find a...