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.

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Q: Surface of a sphere is a two-dimensional Riemannian space. Compute the Chrtstoffel symbols.

A: Given: Surface of a sphere is a two-dimensional Riemannian space To find: Chrtstoffel symbols

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Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Let a E R" be fixed. Suppose that vectors x, y E R" are related by the equation a = x+ (x.y)y. a)…

A: Solutions

Q: Q3/(a) Let d be a trivial metric on a set of X, find the induced topology on X by a metric d. (b)…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and ВС (Х, М) :%3D…

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Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

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Q: <x tet fecl [a,b] and oege have we = f(n) ganze x Je count + fit-LED Plant at y - (x-[x]) f(x) +…

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

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Q: Let X={2,3,4,.} and A= {3,4,5} and metric defined on |X as d(x,y) = |x - y Give the last answer Find…

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Q: Using the fact that A/B = An Be, show that in a metric space, bdy(A) = bdy (Aº)

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Q: Use the inner product (f, 9) = | f(x)g(x) dr %3D in the vector space CU[0, 1] of continuous…

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Q: (a) A¹\{1} (b) V(x+x) CA² (c) V(x2-x7,x3-x})\{0} CA³

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Q: M is an infinite set. If x,y ∈ M, explain the relation f below            {  1 , if x = y   f(x,y)…

A: We have to solve given problem:

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Q: Is it possible to have a function on a metric space which is discontimuous at every point of the…

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Q: Let C[0, 1] be the vector space of continuous functions that are defined on the interval [0, 1].…

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Q: Let X=(7,8,9,. and A-(8,9) and metric defined on X as d(x.y) |= |x -y Give the last answer Find…

A: The given set is: X=7, 8, 9, ⋯ and A=8, 9 The metric d is defined on X, as follows, dx, y=x-y This…

Q: If y and z are functions of u and v, then the Jacobian of the function transforming (x, Y, z) → (x,…

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Q: Give an example that satisfies in arbitrary metric space, boundedness and closedness do not…

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Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: dual of c, is isometric to l'.

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Q: Which among the following diagrams correspond to the metric below on the real plane R^2?…

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Q: Theorem (Axioms of a Derived set): Let (M,d) be a metric space and let S,T C M. Then (1) S CT S' S…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: Let X={5,6,7,...} and A={5,6,7} and metric defined on X as d(x,y) |= |x -y Give the last answer Find…

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Q: (b) 22). Compute the partial derivative frryz for f(x, y, z) = cos(cos(x³ + z²)) + x² In(y² +

A: Given f(x,y,z)=cos(cos(x3+z2))+x2ln(y2+z2).We need to find the partial derivative fxxyz for the…