As you were asked to show in Question 5, the equation r^ (x, y) = (rx, y) defines a group action of (R+,x) on the plane R². Determine the orbit of each of the following points: (i) (1,0), (0, 1), (-1,1). (ii) Describe all the orbits of the action geometrically. (iii) Determine the stabiliser of each of the following points: (1,0), (0,1). tange (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying one error or significant omission in each part. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include statements or omissions that follow logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) (i) r^(1,0) = (r, 0), so Orb(1,0) = {(r,0): r ER}. TA (0, 1) = (0, 1), so Orb(0, 1) = {(0,1)}. r^(-1,1)= (-r, 1), so Orb(-1, 1) = {(-r, 1) : r ER}. (ii) Orb(1,0) is the line y = 0, the x-axis. Orb(0, 1) is the single point {(0, 1)}. Orb(-1,1) is the horizontal line y = 1. (iii) r ^ (1,0) = (r,0) = (1,0) if and only if r = 1, so Stab(1,0) = {1}. r^ (0, 1) = (0, 1) for all values of r, so Stab(0, 1) = {(0,1)}. Write out a correct solution to the exercise.
As you were asked to show in Question 5, the equation r^ (x, y) = (rx, y) defines a group action of (R+,x) on the plane R². Determine the orbit of each of the following points: (i) (1,0), (0, 1), (-1,1). (ii) Describe all the orbits of the action geometrically. (iii) Determine the stabiliser of each of the following points: (1,0), (0,1). tange (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying one error or significant omission in each part. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include statements or omissions that follow logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) (i) r^(1,0) = (r, 0), so Orb(1,0) = {(r,0): r ER}. TA (0, 1) = (0, 1), so Orb(0, 1) = {(0,1)}. r^(-1,1)= (-r, 1), so Orb(-1, 1) = {(-r, 1) : r ER}. (ii) Orb(1,0) is the line y = 0, the x-axis. Orb(0, 1) is the single point {(0, 1)}. Orb(-1,1) is the horizontal line y = 1. (iii) r ^ (1,0) = (r,0) = (1,0) if and only if r = 1, so Stab(1,0) = {1}. r^ (0, 1) = (0, 1) for all values of r, so Stab(0, 1) = {(0,1)}. Write out a correct solution to the exercise.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:This question concerns the following exercise.
(b)
Exercise
As you were asked to show in Question 5, the equation
r^ (x, y) = (rx, y) defines a group action of (R+,x) on the plane R².s dos
(i) Determine the orbit of each of the following points:
(1,0), (0,1), (-1,1).
clos
(ii) Describe all the orbits of the action geometrically.
(iii) Determine the stabiliser of each of the following points:
(1,0), (0, 1).
(a) Explain why the following solution to this exercise is incorrect and/or
incomplete, identifying one error or significant omission in each part.
For each error or omission, explain the mistake that the writer of the
solution has made. (There may be more than three errors or omissions,
but you need identify only three. These should not include statements
or omissions that follow logically from earlier errors or omissions.)
Solution (incorrect and/or incomplete!)
(i) r^(1,0) = (r, 0), so Orb(1,0) = {(r, 0): r ≤ R}.
r ^ (0, 1) = (0, 1), so Orb(0, 1) = {(0,1)}.
r^(-1, 1) = (-r, 1), so Orb(-1, 1) = {(-r, 1) : r ≤ R}.
(ii) Orb(1,0) is the line y = 0, the x-axis.
Orb(0, 1) is the single point {(0,1)}.
Orb(-1,1) is the horizontal line y = 1.
(iii) r ^ (1,0) = (r,0) = (1,0) if and only if r = 1, so Stab(1,0) = {1}.
r^ (0, 1) = (0, 1) for all values of r, so Stab(0, 1) = {(0,1)}.
Write out a correct solution to the exercise.
(31
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