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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

    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)

    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)

    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

    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

    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

    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.

    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 an odd function is strictly increasing

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

    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...
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    Monotonicity of odd function and 1-1 function

    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...
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    Find functions with given domain and range

    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...
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