dynamical systems

sachmo

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Nov 15, 2004
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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
 
YOU MUST CLARIFY THIS QUESTION!

It asked about sets being opened, closed, or neither.
Is that algebraic, that is closed with respect to an operation. If so, what operation?
Or, is it a topological property? If so, what is the topology? The standard metric on ℜ?

Also, WHAT IF THE FUNCTIONS B(x) & C(x)???
 
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