Friday, February 20, 2009
Correction answer problem sheet 4
In question 5 the solution x(t)=t^{3/2} should be replaced by x(t)=(2t/3)^{3/2}. It has been corrected on the solutions sheet.
Tuesday, February 17, 2009
Course notes chap 5
Please find temporary versions of the course notes here and of slides here. As I am writing them at the moment, please be prepared for corrected updates (hopefully) soon to appear in the left side margin of this website. In the mean time please refer to the lecture notes (written).
Sunday, February 15, 2009
Problem sheet 3 corrections
The last entry in the spherical coordinate transformation should be "rcos(theta)" instead of "cos(theta)". Thanks to the alert student who pointed this typo out to me.
Furthermore, in question 3 it should have been mentioned that F is continuous, and in question 4 the formula to be derived was missing a derivative: "(F-1)' " instead of "F-1".
In question 8 I clarified/reformulated the meaning of "F-G" in the question and its answer.
I have made the corresponding corrections to the problem sheet.
ps: I also added a "t>=0" to the last part of question 3 on Sheet 2.
Furthermore, in question 3 it should have been mentioned that F is continuous, and in question 4 the formula to be derived was missing a derivative: "(F-1)' " instead of "F-1".
In question 8 I clarified/reformulated the meaning of "F-G" in the question and its answer.
I have made the corresponding corrections to the problem sheet.
ps: I also added a "t>=0" to the last part of question 3 on Sheet 2.
Thursday, February 12, 2009
12/2 lecture: correction of argument
During the lecture I gave an incomplete argument why the inequality |T(u(t))-T(v(t))|<= Ka d(u,v) implies that T is a contraction if aK<1. (As correctly pointed out to me by one of you.)
Clearly, d(T(u),T(v))>=|T(u(t))-T(v(t))| for any t, but this observation does not lead to the required conclusion, since the inequality is in the wrong direction.
An argument that leads to the desired result is that since |T(u(t))-T(v(t))| <= Ka d(u,v) FOR ALL t in J, we have in particular d(Tu,Tv)=sup_t|T(u(t))-T(v(t))|<= Ka d(u,v).
Clearly, d(T(u),T(v))>=|T(u(t))-T(v(t))| for any t, but this observation does not lead to the required conclusion, since the inequality is in the wrong direction.
An argument that leads to the desired result is that since |T(u(t))-T(v(t))|
Monday, February 9, 2009
On the definition of asymptotic stability
One of you asked me how the fact that "all solutions with initial conditions in a neighbourhood of an equilibrium converging to an equilibrium as time goes to infinity" (as in the definition of asymptotic stability) should imply Lyapunov stability. This is a good question, as it highlights an important fact. Asymptotic stability of an equilibrium solution is a local concept, requiring first of all the equilibrium to be Lyapunov stable (read carefully the given definition of asymptotic stability). It is namely possible that all solutions with initial conditions in the neighbourhood of an equilibrium converge to the equilibrium as time goes to infinity, but that the equilibrium is not Lyapunov stable. It may happen that some solutions converge to the equilibrium but only by wandering first far away from the equilibrium.
As an example, consider the flow of a vector field on the circle, dx/dt=1+sin(x) (with x in [0,2pi)). This flow has one equilibrium, x=-pi/2, and all solutions converge to this equilibrium as time goes to infinity. However, if one starts very closely on one side of the equilibrium, the solution may take you all the way around the circle to the other side, since all solutions converge to the equilibrium from the same side!
As an example, consider the flow of a vector field on the circle, dx/dt=1+sin(x) (with x in [0,2pi)). This flow has one equilibrium, x=-pi/2, and all solutions converge to this equilibrium as time goes to infinity. However, if one starts very closely on one side of the equilibrium, the solution may take you all the way around the circle to the other side, since all solutions converge to the equilibrium from the same side!
Progress test
Please note that the progress test and model answer are now available from the left margin.
(I was pointed to a typo in d(ii) which has now been corrected: x(0)=(3,0)^T , and not (1,0)^T.)
(I was pointed to a typo in d(ii) which has now been corrected: x(0)=(3,0)^T , and not (1,0)^T.)
Friday, February 6, 2009
Answers to question 1(c) Sheet 2 and 3(a,b) Sheet 1
Someone alerted me to the fact that there is a mistake in this answer. The method is in principle ok, but after the change of coordinates that leads to the decoupling of the ODE it is claimed that a simple ODE for z has a certain simple solution. In this solution the variables a and b appear slightly wrongly: the first two terms should be -bt/a-b/a^2 (so a and b should be interchanged). This has now been corrected.
There is also a typo in the answer of questions 3(a,b) of sheet 1. The answers are correct but in both parts we have A=PJP^{-1}. This has now been corrected.
There is also a typo in the answer of questions 3(a,b) of sheet 1. The answers are correct but in both parts we have A=PJP^{-1}. This has now been corrected.
Thursday, February 5, 2009
Chapter 2 lecture notes correction
I revised the last part of section 2.3.2 of the lecture notes (Chapter 2), p19, as it lacked care in the details (and as written contained some mistakes). The current version is correct and extensive detail is given. Thanks to all of you who pointed out to me that this part of the notes was hard to understand! (keep up the good work :-))
In addition, in relation to a question by another one of you, I decided to add a line to the proof of prop 2.5.1 to recall that F(x(t),y(t))=0 [as this is used].
In addition, in relation to a question by another one of you, I decided to add a line to the proof of prop 2.5.1 to recall that F(x(t),y(t))=0 [as this is used].
Tuesday, February 3, 2009
No lecture on Feb 3
Apologies for the late notice. I suggest you start studying the posted answers to problem sheet 2 (in preparation of next week's progress test).
Monday, February 2, 2009
9 February progress test
The progress test of 9 February will concern the material of chapters 1 and 2 of the course notes,
and the related problems on sheets 1 and 2. Please note that the objective of the test is to see whether you have understood the material and not whether you know it by heart (so please do not waste your time focusing on the latter: for instance, while you are encouraged to work through and understand the proofs of problems 4 and 6 on sheet 2, please do not learn such proofs by heart!).
and the related problems on sheets 1 and 2. Please note that the objective of the test is to see whether you have understood the material and not whether you know it by heart (so please do not waste your time focusing on the latter: for instance, while you are encouraged to work through and understand the proofs of problems 4 and 6 on sheet 2, please do not learn such proofs by heart!).
Sunday, February 1, 2009
Updates
In reaction to comments and question from students (you!), I have made a few changes to the text of Chapter 2: discussion at end of par 2.3.2 on p19 and end of proof of proposition 2.4.2 on p22. I also added a few notes on inner products on complex vector spaces (see left margin). I have also added some more detail to the solutions of Q3 and Q4 of problem sheet 1. Model answers for problem sheet 2 have also been posted.
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