1. Carefully write the following:
(a) Suppose A is a 3 × 3 matrix that you can diagonalised, explain how you would diagonalise A. (1 mark)
(b) Give an example of two unbounded functions f : (−1, 1) → R and g : (−1, 1) → R such that f + g is bounded and L-Lipschitz for every L > 0. (1 mark)
(c) The definition of sup(A) and the definition of f : (0, T) × Ω → R being bounded and being L-Lipschitz in the second variable. (1 mark)
(d) Explain why the Picard iterates satisfy x1 = xk for all k > 2 if we consider an IVP for x ∈ C1 ([0, T]; R) with the DE x'(t) = g(t) for some continuous function g : (0, T) → R. (1 mark)
Part (d) please
Given that
we have to
1. Carefully write the following:
(a) Suppose A is a 3 × 3 matrix that you can diagonalised, explain how you would diagonalise A. (1 mark)
(b) Give an example of two unbounded functions f : (−1, 1) → R and g : (−1, 1) → R such that f + g is bounded and L-Lipschitz for every L > 0. (1 mark)
(c) The definition of sup(A) and the definition of f : (0, T) × Ω → R being bounded and being L-Lipschitz in the second variable. (1 mark)
(d) Explain why the Picard iterates satisfy x1 = xk for all k > 2 if we consider an IVP for x ∈ C1 ([0, T]; R) with the DE x'(t) = g(t) for some continuous function g : (0, T) → R. (1 mark)
1. Carefully write the following: (a) Suppose A is a 3 × 3 matrix that you can diagonalised, expl...
1. Carefully write the following: (a) Suppose A is a 3 × 3 matrix that you can diagonalised, explain how you would diagonalise A. (1 mark) (b) Give an example of two unbounded functions f : (−1, 1) → R and g : (−1, 1) → R such that f + g is bounded and L-Lipschitz for every L > 0. (1 mark) (c) The definition of sup(A) and the definition of f : (0, T) × Ω → R...
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
Problem 11.11 I have included a picture of the question (and the referenced problem 11.5), followed by definitions and theorems so you're able to use this books particular language. The information I include ranges from basic definitions to the fundamental theorems of calculus. Problem 11.11. Show, if f : [0,1] → R is bounded and the lower integral of f is positive, then there is an open interval on which f > 0. (Compare with problems 11.5 above and Problem...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
Exercise 3.1.12: Prove Proposition 3.1.17. Exercise 3.1.13: Suppose SCR and c is a cluster point of S. Suppose : S R is bounded. Show that there exists a sequence {x} with X, ES\{c} and lim X e such thar S(x)} converges. and g such thal 2 2 asli and 8 ) Las y C2, bulg 1)) does not go lo L as is, find x → Exercise 3.1.15: Show that the condition of being a cluster point is necessary to...
Both part of the question is True or False. Thank you Problem 1. (ref. Example 3 in the slide) Let X = Y = C[0, 1] (with the norm || ||C[0,1] = sup |u(x)]). For any u € C[0, 1], define T€[0,1] v(t) = u(s)ds. We denote by T the mapping from u to v with D(T) = C[0, 1], i.e., v(t) = Tu(t). Then, the following conditions are true or not? Example 3. We denote by the set of...
Problem 3. (1) Let H be a Hilbert space and S, TE B(HH). Then, prove that ||ST|| ||||||||| (2) Let X, Y be Hilbert spaces and Te B(X,Y). Then, prove that ||1||| sup ||T3|1 TEX=1 Let X, Y be Banach spaces. Definition (review) We denote by B(X, Y) a set of all bounded linear operators T:X + Y with D(T) = X. B(X, Y) is a vector space. Definition (review) A linear operator T:X + Y is said to be...