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Math Help - Dynamical Systems

  1. #1
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    Dynamical Systems

    Q!. Decide whether each of the following func are 1-1,onto,homeomorphisms or diffeomorphisms
    a. f(x)=X to the power5/3
    b. f(x)=X to the power4/3
    c. f(x)=3x+5
    d. F(x)=e to the power x
    e. f(x)=1/x
    f. f(x)=1/x to the power 2

    Q2. Identify which of the following subsets of R are closed,open or neither
    a. (x,where x is an interger)
    b. (x,where x is a rational number)
    c. (x,where x=1/n for some natural number n)
    d. (x,sin(1/x)=0)
    e. (x,xsin(1/x)=0)
    f. (x,sin(1/x)>0)

    Q3. Sketch the graph of B(x)

    Q4. Prove that B'(0)=0

    Q5. Inductively prove that B^n(0)=0 for all n.Conclude that B(x) is a c infiniti function.

    Q6. modify B(x) to construct a C infiniti function C(x) whcih satisfies
    a. C(x) =0 if x is less than or equal to 0
    b. C(x) =1 if x is greater that or equal to 1
    c. C'(x)>0 if 0<x<1

    Q7. Modify C(x) to construct a C infiniti bump function D(x) on the interval [a,b], where D(x) satisfies
    a. D(x) =1 for a<x<b
    b. D(x) = 0 for x<alpha and x>beta where a<alpha and beta>b
    c. D'(x) not equal 0 on the intervals (alpha,a) and (b,beta)

    Q8. Use a bump function to construct a diffeomorphism f;[a,b] goes to [c,d] which satisfies f'(a)=f'(b)=1 and f(a) =c,f(b)=d

    Please any kind of help would be appreciated.
    SK
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  2. #2
    hpe
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    Quote Originally Posted by sachmo
    Q!. Decide whether each of the following func are 1-1,onto,homeomorphisms or diffeomorphisms
    a. f(x)=X to the power5/3
    b. f(x)=X to the power4/3
    c. f(x)=3x+5
    d. F(x)=e to the power x
    e. f(x)=1/x
    f. f(x)=1/x to the power 2
    a) homeo from R to R, 1-1 and onto, not diffeo
    b) none of the above
    c) diffeo from R to R, 1-1 and onto
    d) diffeo from R to R+, 1-1
    e), f) none of the above

    You still should convince yourself of every single assertion and give a proof.
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