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- 1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {phi_t}) be a flow on a metric space x. what is x0 in x a fixed point of the flow? c) When do you say that a fixed point x0 in x is Poincare stable? d) When do you say that a fixed point x0 is Lyapunov stable? Use Analysis to complete the following statements.Let X be a set, and define d(x,y) ={0, if x=y; 1, otherwise (a) Show that the function d defines a metric on X. (b) A sequence (xn) ⊆ X is called eventually constant if there is N ∈ N such that xm=xn; m, n > N. Show that any eventually constant sequence converges. (c) Show that, if a sequence (xn) is convergent under the discrete metric, then (xn) must be eventually constant?1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?
- 1. Use the definition of the limit ( epsolon - delta ) to show thatlim of 1/z as z approaches -i2. Give the condition which ensure that |ez| < 1 where z in C.Evaluate the limit limn→∞ ∑ n i=1 f(ci ) Δxi over the region bounded by the graphs of the equations.Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function defined on R, such that for every x it holds that |a(x)| ≤ 1. Using the Global Picard–Lindel¨of Theorem, show that there exists a unique solution y defined on R.
- let {x, {phi t}} be a flow on a metric space x. when is x0 is in x a fixed point of the flow when do you say that a fixed point x0 in x is Pointcare stable? when do you say that a fixed point x0 is Lypunov stable?Consider the interval [0, 1]. (a) If δ(x) = 1/9, find a δ(x)-fine tagged partition of [0, 1]. Does the choice of tags matter in this case? (b) Let δ(x) ={ 1/4 ifx = 0 / x/3 if0< x ≤ 1. Construct a δ(x)-fine tagged partition of [0,1]. The tinkering required in Exercise 8.1.6 (b) may cast doubt on whether an arbitrary gauge always admits a δ(x)-fine partition. However, it is not too difficult to show that this is indeed the case.Let x denote the amount of time for which a book on 2-hour reserve at a college library is checked out by a student, and suppose that x has density func- tion f(x)5.5x for 0,x,2.
- Prove that the d1 (“Manhattan”) metric on R2 is really a metric.Prove that limit of x^4cos2/x=0 , as x approaches zero.We are building an oblique tower from n pieces of identical, homogeneous blocks on a horizontal surface, according to the figure. What is the maximal possible distance d by which the topmost block is shifted horizontally relative to the block at the bottom? Describe the positions of the blocks in this extreme case and determine the function d(n). What is the limit of d(n) as n → ∞? (The blocks have unit length.)