MS3011: Week 12

March 27, 2013 in MS 3011 | 12 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.


butterfly

Review Tutorial/Lecture

Wednesday 1 May at 10:00 in WGB G014.

Feedback

Thank you for your feedback today.

I would make the following comments:

The Homework is hard but fair: ye are final year students. Also I have repeatedly said that I am willing to answer questions about it.

It has everything to do with what we are doing in class (iterator functions, fixed points, orbits, etc.) and ye are supposed to know about the other topics from other modules. 

I agree that it might require a lot of thought for 12.5% but when you are finished with it I have no doubt whatsoever that your understanding of the material can only be increased.

More exercise sheets? Agreed — although I didn’t see much evidence of us doing the too-few questions that I was posing weekly.

Regarding getting your tests back: ye have an option to view them but I need to keep them I’m afraid.

Tutorials

Summer 2012: Question 4 (e),  (f)

Autumn 2012: Question 4 (d), (f)

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question on the Tent Mapping.

Week 12

We summarised our work on roots of unity and illustrated the following result… parts 4 & 5 are not examinable so don’t worry that it wasn’t covered in lectures. In fact as it turned out the proof was a lot harder than I thought… it is not true for all \alpha — only for almost all irrational \alpha when f(z)=z^2 as far as I know ! The following proposition is true though.

Proposition

Suppose that f:\mathbb{C}\rightarrow\mathbb{Z} is a power mapping

f(z)=z^n for some n\geq 2.

Then the dynamical system (\mathbb{C},d) exhibits the following behaviours:

  1. If z_0\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\} then z_n\rightarrow 0.
  2. If z_0\in\mathbb{C}\backslash\bar{\mathbb{D}}:=\{z\in\mathbb{Z}:|z|>1\} then z_n\rightarrow\infty
  3. If z_0=e^{q\,2\pi i} with q\in\mathbb{Q} then z_0 is eventually periodic.

Suppose now that f(z)=z^2. Then

4.    If z_0=e^{\alpha\,2\pi i} with \alpha\in\mathbb{R}\backslash\mathbb{Q} for, in binary,

\alpha=0.0101100011000001010100110101011111\dots_2

Then z_0 has a dense orbit.

(In fact, (\mathbb{T},z^2) is a chaotic mapping.)

     5.     Suppose now that n>2  and we are again looking at f(z)=z^n. Then there exists an \beta\in\mathbb{R}\backslash\mathbb{Q} such that the orbit of z_0=e^{\beta\cdot 2\pi i} is not dense in \mathbb{T}.

Read the rest of this entry »

MATH7021: Weeks 7 & 8

March 23, 2013 in MATH7021 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test 2

Having spoken with Ted, Test 2 — on Laplace Methods — will now be held on Monday 29 April in Week 12 (rather than Week 11). I have a sample ready here — for q.1(a) let x'(0)=0.

Weeks 7 & 8

Plenty of Laplace

Week 9

Finish off Laplace. I will give ye the Laplace-Mega-Sheet (TM) also.

More Explanations

Somebody told me the beauty of learning about maths off a video is that you can pause the lecturer. I found these online and I have to say I like these lectures a lot… not many examples but we do them in class even if you seem to think that I don’t!

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/

Ye would have covered lectures 11 to 14 last semester.

Lectures 19 & 20 are relevant for us. Lecture 22 and 23 are where this stuff ends up being really useful (not in MATH7021 unfortunately).

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question on an inverse Laplace transform.

Weeks 10 – 12

Multiple Integration.

Tutorials

Yes the following table should be A-L and M-Z not A-L and K-Z!

tutorials

MS3011: Weeks 10 & 11

March 23, 2013 in MS 3011 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Weeks 10 & 11

We spoke about complex number multiplication — particularly the geometric aspect including DeMoivre’s Theorem and Roots of Unity. A good summary of what we covered may be found here actually.

We also said that almost all of our dynamical systems theory carries over to when when we have a complex-valued function

f:\mathbb{C}\rightarrow \mathbb{C},

rather than a real-valued function f:\mathbb{R}\rightarrow \mathbb{R}. We will look at this in more detail next week, our final week.

Tutorials

Good questions to look at (some of these have been looked at in tutorials):

Summer 2009 Question 5

Autumn 2009: Question 5

Summer 2008: Question 5

Summer 2012: Question 4 (a) – (e)

Autumn 2012: Question 4 (a) – (d)

Summer 2010: Question 4 (b)

Autumn 2010: Question 3 (a), 4 (a)

Summer 2009: Question 1 (c), (d), 2 (c)

Autumn 2009: Question 1 (c), (d), 2 (c)

Summer 2008: Question 1 (c), (d), 2 (c),

MATH6040: Weeks 7 & 8

March 23, 2013 in MATH6040 | 2 comments

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Weeks 7 & 8

In Weeks 7 & 8 we began our study of some further topics in differentiation: namely parametric differentiation, implicit differentiation and partial differentiation.

Week 9

In Week 9 we must finish our study of partial differentiation — look at its applications to error analysis — and talk about related rates (which I probably should have done first but anyway!)

Test 2

We will have test 2 on the entirety of the further differentiation material in Week 11. Please find a sample. This is a tough test: ye have enough to do questions one to three. By the end of Week 9 — or at worst the start of Week 10 you should be able to do all of the questions. The test will most likely be held on Wednesday April 23.

Notes

A relatively recent set of lecture notes — including partial differentiation. Implicit Differentiation notes and exercises are here also…also you might find this E-book useful.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about parametric differentiation.

MATH6037 Weeks 7 & 8

March 23, 2013 in MATH6037 | 3 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test Results

You are identified by the last five digits of your student number.

S/N Mark
08272 93
64673 90
91054 78
50902 70
79455 65
92344 61
92352 58
64674 48
93152 40
58171 30
92343 28
04346 28
92345 25
93150 25
92351 15
51215 0

Revision

Please feel free to ask me questions via email or even better on this webpage — especially those of us who struggled in the test.

The worked solutions of all the Winter 2012 stuff we covered is here.

Please find a reference for some of the prerequisite material here.

Week 7

In Week 7 we looked at approximate integration. The notes that I gave you had a slightly different notation to the one used in the mathematical tables. Here are these notes except in a notation that is consistent with the mathematical tables. A corollary of this is that there will be no questions on errors in approximate integration.

Week 8

In Week 8 we did Euler’s method and we also did some Newton-Raphson and indeed Euler in the Maple Lab.

Weeks 9 – 12

Eight and half good hours of Laplace Methods — a Maple Lab and an open book Maple test. I will send you a sample Maple test early this week.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this rather technical question on the Euler Method.

MATH6040: Week 6

March 7, 2013 in MATH6040 | Leave a comment

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test Results

You are identified by the last five digits of your student number.

Course S/N %
BIS 90235 97
BIS 91290 93
BIO 90070 93
BIO 87815 87
BIO 89394 83
BIO 88030 73
BIO 82233 73
BIO 70254 73
BIS 79030 70
BIO 82806 70
BIS 91289 63
BIO 80367 60
BIO 89841 57
BIO 80366 57
BIO 81497 57
BIS 42888 53
BIS 44485 53
BIO 67933 50
BIO 75669 43
BIO 88610 43
BIO 74330 37
BIO 85561 37
BIO 86561 37
BIO 82783 33
BIO 89290 30
BIO 87718 27
BIO 05843 23
BIS 74812 17
BIO 88130 17
BIO 89114 10
BIS 56771 –
BIS 73953 –
BIO 75961 –
BIO 86247 –
BIO 72263 –

Week 6

In Week 6 we finished off vector algebra. Hopefully you won’t forget this image:

tolietrollvector

Note that the cross product of two vectors is perpendicular to both of the vectors — the same way the toilet roll holder is perpendicular to both of my arms here… the ‘toilet roll vector’ is in the same direction as \mathbf{u}\times \mathbf{v} in this picture:

tolietrollvectorone

Week 7

In Week 7 we will begin our study of some further topics in differentiation: namely related rates, parametric differentiation, implicit differentiation and partial differentiation.

Notes

A relatively recent set of lecture notes, with gaps.

MATH7021: Week 6

March 7, 2013 in MATH7021 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test Results

You are identified by the last five digits of your student number. Two students never wrote down their name nor student number and should make themselves known to me.

S/N Test 1 %
51004 100
45948 98
49905 98
68681 97
43371 95  *Not Registered?
93761 90
49846 82
Unknown 1 81
93763 76
73685 76
74238 76
61502 76
72110 76
Unknown 2 76
74033 71
74049 65
72743 61
70398 55
59158 55
73289 52
61788 50
61939 47
77537 47
46581 45
70213 44
59616 42
71112 40
67367 39
66644 39
74242 29
58461 24 *Not Registered?
72127 23
56925 21
60829 21
61941 18
74564 16
69699 13
60831 –
76010 –
17751 –
70110 –
44985 –
72983 –
44439 –
61904 –
46134 –
56695 –
22204 –
15818 –

Week 6

We defined what the Laplace transform is and found the Laplace transforms of some common functions. We proved that the Laplace transform is linear and this allowed us to find the Laplace transform of sine and cosine functions via the Euler Formula. We also proved the First Shift Theorem.

Week 7

Next week we will work slowly and deliberately developing our Laplace stuff in particular how well it behaves with respect to differential equations.

Tutorials

This week coming it is

13:30 Groups S1 and S3 (M-Z)

14:30 Groups S2 and S3 (A-L)

Yes the following table should be A-L and M-Z not A-L and K-Z!

tutorials

MS3011: Week 9

March 7, 2013 in MS 3011 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test Results

More remarks on these later in the week. Last five digits of student number given.

S/N % Remark
05669 abs S/N
34673 abs SN
01947 abs *
69001 abs S/N
27197 *
34159 100
12147 100
23747 100
80059 100
21209 100
63301 100
44855 100
18159 100
47109 96
86607 96
16233 96
01701 96
79917 92
64454 92
33245 92
12332 88
37258 84
45109 84
75251 80
70869 76
60663 72
69415 72
19259 68
69738 68
15321 64
18577 64
00153 64
54745 60
84229 60
02929 60
21025 56
07705 52
69423 52
07784 48
48443 48
69571 48
Unknown 44
45095 36
40067 36
21931 28
80697 24
78026 24
60543 24
7679 16 not regisd.
06454 0 abs
05587 0 abs
25527 0 abs
67327 0 abs
80133 0 abs
32430 0 abs
70492 0 abs
10684 0 abs

Tutorial Venue

For the rest of term we are in WGB G03

Week 9

We will finished off our work on the doubling mapping and we began talking about complex numbers.

The marking scheme for Summer 2012 Q. 2 is here and the third page of here. All of the Summer 2012 marking scheme may be found here. As I thought, answering Q. 2 (a), (b), (c) and recognising that they suggested that the Doubling Mapping had, respectively, sensitivity to initial conditions, that the periodic points are dense and a point with a dense orbit was good for 16/25 = 72% > 70% of the marks for that question.

Week 10

This week we will begin to talk about the arithmetic of complex numbers — in particular what it looks like.

Tutorials

There aren’t really any questions that we can do this week that we couldn’t do last week. Perhaps you should spend some time looking at the homework and decide which option you want to take.

MATH6037: Week 6

March 7, 2013 in MATH6037 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test

The test will be held next week 13 March. Please find a sample attached (you will also be given these tables although you can use your own tables also if they are clean). This sample is not a guide to the questions or their marks but only the layout of the test:

  • Answer all questions
  • 1 hour test 19:10 – 20:10
  • Q.1 Integration by Parts, Q. 2 Partial Fractions Q.3 Partial Differentiation Q. 4 Differentials and Error Analysis Q. 5 Root Approximation

We will have the test from 19:10 – 20:10. After the break we will start a proper look at approximate integration.

There are some students with a very busy Week 7: those students with physics or a heavily pregnant spouse. Ye will have an opportunity to sit the test at 20:15 on Thursday 21 March which is in Week 8. Ideally you should email me if you fall into this bracket. If you are sitting the test in Week 8 you don’t need to turn up next Wednesday until 20:30.

Revision

Please feel free to ask me questions via email or even better on this webpage.

Week 6

In Week 6 we did some revision, namely the sample test. After this we used Maple to help us revise partial differentiation.

Week 7

Test and then a proper look at approximate integration.

Week 8

Finish off numerical methods and maybe start Laplace methods. We will do some numerical methods stuff in Maple.

MATH6040: Week 5

March 1, 2013 in MATH6040 | Leave a comment

I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Test 1

The test will be held this week Wednesday 6 March). Please find a sample. The sample is to give you an idea of the format and length.

Week 5

In Week 4 we will continued our study of Vector Algebra including the Cross Product. We saw that vector algebra has applications to mechanics such as the fact that work can be written W=\mathbf{d}\cdot\mathbf{F} and torque/moment of a force is defined as \tau=\mathbf{d}\times\mathbf{F}.

Week 6

In Week 5 we will finish off vector algebra, have our test and start a review of MATH6015 material: differentiation.

Test 2

Ye will get two weeks of notice for test 2 and it will be on differentiation.

Notes

A relatively recent of lecture notes, with gaps.