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    Solving Nonlinear System using Matlab

    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...
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    How to construct nonlinear ODE systems with given condition?

    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...
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    H is a subgroup of its Centralizer iff H is Abelian

    Yeah, that's true, as simple as it is. Doesn't even need any algebraic proof, just simply explain it out then.
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    H is a subgroup of its Centralizer iff H is Abelian

    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...
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    H is a subgroup of its Centralizer iff H is Abelian

    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...
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    How to find the order of a matrix?

    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...
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    How to count the total # of non-invertible 2x2 matrices

    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...
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    Prove # of m-cycles in Sn (symmetric group)

    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...
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    Prove # of m-cycles in Sn (symmetric group)

    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...
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    Find the order of the cyclic subgroup of D2n generated by r

    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...
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    Prove t = x*y => t*x = x*t^(-1)

    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!
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    Prove t = x*y => t*x = x*t^(-1)

    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...
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    Prove x^2=1 if and only if the order of x is 1 or 2

    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?
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    Prove x^2=1 if and only if the order of x is 1 or 2

    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)...
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    Need example of a continuous function map cauchy sequence to non-cauchy sequence

    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...
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    A is a root of order of polynomial p iff p(a)=p'(a)= =[p^(k-1)](a)=0

    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...
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    A is a root of order of polynomial p iff p(a)=p'(a)= =[p^(k-1)](a)=0

    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) :=...
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    Darboux integration, show inequality

    Yeah, I thought so too, just need second opinion. Thanks!
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    Darboux integration, show inequality

    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...
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    Darboux integration, show inequality

    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...
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    Prove Lower Integral <= 0 <= Upper Integral

    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...
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    Isolated Point

    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...
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    Show set has no limit points.

    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.
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    Show set has no limit points.

    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...
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    Prove that f(a) < f(c) < f(b)

    Thanks so much for the clarification fzero!
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    Prove that f(a) < f(c) < f(b)

    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)...
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    Prove that f(a) < f(c) < f(b)

    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...
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