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Questions, comments and answers.
14 October 2009
Schol I was wondering if this year's maths schols paper or at least your questions in the paper will include any of the JF maths. Thank you
Reply No, it will only be based on material from this year,
18 October 2009
PS2 Q3 I have a question about last weeks tutorial. Why in part 3 do you put n=2m+1
Reply
So sin \pi n/2 is quite difficult, for n even it gives zero, for n odd it alternates
1 for n=1,5,9,13, . . .
-1 for n=3,7,11, . . .
To see this just plot out sine.
The easiest way to deal with this is to change to n=2m+1; then
n=1 <-> m=0, n=5 <-> m=2, n=9 <-> m=4, ie even values of m
n=3 <-> m=1, n=7 <-> m=3, etc, odd values of m
so
sin \pi (2m+1)/2 =(-1)^m
which is perhaps the easiest way to express the funny behaviour of the sine.
More generally, when you have a sum over odd numbers, it is often useful to change to 2m+1 since you get all the odd numbers for m a natural number, similarly, if you have a sum over even numbers, it is often useful to change to 2m.
22 October 2009
PS3 Today's tutorial was really long. It wasn't necessarily difficult but the questions required some time to be finished. I usually finish all the tutorial questions in time but it was not the case today. I barely finished the first and the second question. I probably could have finished the problem sheet in 1 hour and a half but not in 50 min. Can you please provide hints or help with long questions to make them more approachable in the given time. Thank you
Reply Yes, it turned out longer than I had indended, I thought the similarity to the example in class would make it quicker. I will try and make next week's sheet shorter.
18 October 2009
Schols Hello,I will be sitting schols this year and I was wondering whether you could tell me if there will be both proofs and applications, ie examples and questions, in your part of the paper, or if I should concentrate on one aspect. Thank you for your time
Reply The emphasis will be on examples, but there will be some theory too;calculating the formula for a_n and b_n in the Fourier series would be a typical example.
24 October 2009
Kronecker delta Hi, I don't really understand when and where you can use the kronecker delta, is there some short explanation you can give or even direct me to a book that might help!?
Reply The Kronecker delta \delta_{mn} is a symbol for one if m=n and zero otherwise; it can be thought of as the (m,n)-entry of a identity matrix. It is not so much a question of when you can use it, but when it arises in a calculation. For example if m is not equal n, then int_{-\pi}^pi exp(i(m-n)t)dt is zero, but when m=n the integral is 2\pi, so we write
int_{-\pi}^\pi exp(i(m-n)t) dt = 2\pi \delta_{mn}
This just means the integral is zero for m and n different and 2\pi when they are the same. The wikipedia article describes this. One of the most significant properties of a kronecker delta is the way it busts out of a sum
\sum_{n=0}^\infty a_n\delta_{nm} =a_m
This is because delta_{nm} is zero when n is not equal to m, so the only term in the sum that is nonzero is the one one where n=m and for that term the delta gives one.
Further message Hi, it was me who asked about the kronecker delta so I just thought it would be polite to reply and tell you that I understand it now, so thanks!! :)
Reply Great.
30 October
solutions Hi Conor, i wanted to look at solutions to problem sheets but it wont let me open them in any format,do i need to download recent adobe or anything?
Reply This is worrying because it works for me; are you clicking on the pdf? It should be very straightforward pdf, so any pdf reader should work. What happens when you click on it?
Further message I can access it from college comp just not from home.it tells me to choose program(adobe etc) to view pdf and then says i have to download something!however i have printed them from college comp so no need to worry...thanks.
Reply Yes, you will need to have a pdf viewer installed, acroread being the obvious one, but there are others: I use evince for example.
4 November 2009
Tutorial timetable Having the three tutorials all on Thursday, and two at the same time, makes it more likely that I'll miss a few.
Reply I am afraid this is beyond my control, I think it makes sense to have the tutorials on the same day, otherwise different classes could end up being take by people who had been to different numbers of lectures, but I agree it is a pity to have them at the same time. Do try not to miss them though, going to the tutorials is the most important thing as far as the exams are concerned.
4 November 2009
Fourier transform of the Gaussian In your lecture today (03/11/09) you mentioned how in the exam you'd either give the result of the "completing the squares" trick or you would guide us through! I was just curious as to whether it would be the same for the schols exam or would we be expected to be able to do that ourselves?
Reply Asking you to work out the fourier transform of exp(-alpha t^2) with no further hints, baring the table of Fourier formulas, would be a reasonable schol question.
15 November 2009
Completing the square. In your completing the squares example, when you got to (alpha t +v)^2 and expanded you arrived at the conclusion that v= -beta/2 sqrt{alpha}. I was just wondering if there was a method to figuring this out or if it was more a case of trial and error or common sense?.
Reply I just compare it to
(at+b)^2=a^2t^2+2abt+b^2
and use that to see what the extra term has to be! So in the example to hand we had
alpha t^2-beta t
so if that is to be a complete square we must have -beta=2 sqrt{alpha} b, hence b=beta/2 sqrt{alpha}
17 November 2009
Piecewise continuous and even and odd I have a question regarding piecewise defined functions. To decide if a piecewise function is even or odd, I usually look at its graph to see its symmetry, but is there a less intuitive way to do this? How can we use the formal definition of even/odd to figure it out? Thank you for your time.
Reply The answer is not much different from looking at it; you just take each piece and ask how it relates to the corresponding piece with the sign of t reversed. Say y=t for t in (-1,1) and 1 for t>1 and -1 for t<-1, then there are just three pieces, the first piece is equal -1 for t<-1, which means -t>1 where y=1 so y(-t)=-y(t), next the y=t bit is clearly odd, and for t>1 y=1 but if t>1 -t<-1 where y=-1. This is an odd function.
19 November 2009
integrating factor Hi Conor,im getting confused with the notation of these ODEs...what is general eqn after multiplying by int. factor? is it y(t)=y(a)e^-I(t)+e^-I(t).INT e^I(z)f(z)?? if so,why is it e^-I(t),where did minus come from? and also the book use different notation... Thanks
Reply So start with the eqn, I will use prime rather than dot for differenciation since it is easier to type: y'+py=f, now work out the integrating factor I(t)=int_a^t p(tau)d tau, multiply across by the integrating factor e^I(t) and we get
e^I(t)y'+e^I(t)py=e^I(t)f(t)
No you can check, the left hand side is in the form of the product rule, so it can be put into product form
[e^I(t)y]'=e^I(t)f(t)
Now integrate both sides
e^I(t)y(t)-y(a)=int_a^t e^I(tau)f(tau)d tau
where I have use I(a)=0 and I changed t to tau before integrating because I want to use t as the variable I am integrating to. To get y(t) we multiply both sides by e^[-I(t)], that's where the minus comes from, I also move the y(a) to the other side
y(t)=y(a)e^[-I(t)]+e^[-I(t)] int_a^t e^I(tau)f(tau)d tau
Does this help?
22 November 2009
Schols Hi Conor, I'm just wondering if anything has changed with regard to you knowing if the scholarship examination paper will require 3 questions to be completed from each of your and Dr. Kitson's course of whether we can choose any 6 from the 8 on the paper. Thank you for your time, and I assume you will post on the Q and A page as a reply but then again to "assume" is to make an "ass" out of "u" and "me"...
Reply Yes; eight questions, four from me, four from Derek, do six with no restriction as to how you divide these between Derek and me.
3 December 2009
Schols will you post the answers to the sample schol paper you gave us
Reply Yes, before the end of term. I am hoping to catch up with stuff next week.
1 December 2009
maths book Hi Conor just wondering if you could recommend a maths book for your course,didnt really find Kreysig that good. Thanking you
Reply Sorry for the delay answering this, the problem is I don't really have an answer, there are lots of books but they are all pretty similar to Kreysig, at least to my mind they have the same virtue, lots of problems, and the same problem; poor explainations and weak intuition. I am hoping to ask around a bit and will get back to you.
30 December 2009
231 PS16 q2 Hello Dr. Houghton, looking back at problem sheet 16 in your 231 course, and the solution that is posted for question 2, I don't see how line (8) follows from line (7) , (in the outline solutions for problem sheet 16). Why is the complementary equation l^2+5l+5=0 , from the previous y''-4y'-5y=e^5z ? I'd appreciate any help.Thank you
Reply You are just being polite and suspect it's a typo! It is a typo: the auxiliary equation should be l^2-4l-5=0; weirdly the solns given for the auxiliary equations were correct even if the equation wasn't! Thanks for pointing this out, it should be correct now.
6 January 2010
Formalas on schol papers Sorry i know this is very late but i was just wondering if there would be general formula's on the schols paper for your part of the course, like there was on the sample paper you gave out?
Reply There will be Fourier formulas like on the sample.
7 January 2010
Frobenius Hello Dr Houghton, just have a doubt about the Frobenius method, when the equation becomes something like ao(stuff)+ a1(stuff)+ indexedsum(stuff)=0, as happens in PS10 and Q4 in the sample schol. Now we take ao arbitrary and get the indicial equation from ao(stuff)=0, but I'm not sure how to deal with the a1(stuff)part: do we need to use it at all? Do we need to check if a1=0 with each r ? or can we set a1=0 and, since ao is arbitrary,ao=1? Sorry for the long question and thank you
Reply This whole thing is tricky and you have to see what works for a given problem. Basically, allowing a_0 to be arbitrary usually forces you to choose r's that will make a_1 zero, however, there are some examples, like the one you mention, where one of the two r's will also allow an arbitrary a_1. However, the bottom line is that there are only ever two solutions to a second order problem, so if you have two r's and hence two solutions with a_0 arbitrary, then the a_1 solution can't be a new solution: it will be equal to one of the solutions you get with a_1 set equal to zero, or some linear combination of them. Hence, it doesn't actually matter what you set a_1 to and you might as well set it to zero.
19 April 2010
Formulas hi dr houghton i was just wondering for the exam MA22S3 will you be giving us the formulas for the fourier series, transform etc or will we learn them? thnak you
Reply There will be Fourier formulas like on the sample.
19 April 2010
Layout Im just enquiring about the layout of the exam, is it 4 q's do 3 or is 6q's do4 like in the sample paper.
For q1b of the sample paper is the answers from the tutorial sheets all that is needed or is a line needed about ao,an and bn
Reply The exam is like the sample: six questions do four and yes, a line would be needed to show how to get a_0, a_n and so on, you multiply across by the the sine or cosine and integrate both sides, the formula for a_n or whatever then pops out using the expression you have just calculated.
19 April 2010
Complex versus trignometric series am just wondering in the solution to the first question on your second schols paper you solve the problem using the formula for Cn, can you do it solving fo Ao, An, Bn too or how do you know when to use which formula?
Reply If you aren't told which to use then either is fine, I find the c_n easier because you can do it all in one go, but they are equivalent and so if the question doesn't say, it doesn't matter which you do.
9 May 2010
Hermite Equation im trying to do some Hermite eqns but im not sure what they are about.
In Problem Sheet 10 Q1,i dont know how you got an answer involving "t"..?
So we start plug in our values for "alpha" and values for n(0,1,2,3....) into given recursion relation and get a1,a2,a3...etc in terms of a1 is it?then figure out what terms will be even and what Will be odd...?
if you get a chance (before exam) could you explain this
thanks
Reply In the question a recursion relation is given for the a_n, so the solution is
y=a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + . . . .
so once you have the a_n you have a way of getting answers involving t's. Now for the Hermite equation the funny thing is that there is an (n-\alpha) in the numerator, so if \alpha is some whole number then there will be an n value where n=\alpha. The recursion relation is two-step, ie it is of the form
a_{n+2} = (stuff) a_n
so n=0 gives a_2 in terms of a_0, n=1 gives a_3 in terms of a_1 etc, hence there is an even series and an odd one, the even series starting with a_0, then a_2, then a_4 and so on; the odd series starting with a_1, then a_3, then a_5 and so on. For alpha odd the odd series will be finite, it will stop when \alpha=n, for even \alpha then the even series stops. Hence, in the question you are looking for the odd series for alpha=3, you work out a_1 and a_3 and then a_5 will be zero.
Let me know if this doesn't help,
9 May 2010
Q2 sample paper i was just wondering for q2, if the period 2pi and are we supposed to integrate over A and -A or t and -t
Reply If its the question I think you mean, it is a Fourier Transform, which means you integrate from -\infty to \infty, but since the function is zero for t>T and t<-T, so you actually integrate from -T to T.
9 May 2010
Sample paper I have been going over the sample paper you gave out and for question 5 you gave the answer (line 14) as y= (C1 +C2t +1/2)expt but should it not be y= (C1 +C2t +1/2t^2)expt as y= Ct^2expt is the anzatz you use and you get C=1/2. If not, why not?
Reply Of course you are right: thanks!