MATH6040: Week 3

February 13, 2014 in MATH6040 | 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.

Important: Gaussian Elimination

Please look at p.26 Questions 1 & 2 and p.35 Questions (iv), (v) and (vi) only. The other ones are just too messy.

Also if you download Maple (see below), there is a Maple Tutor that is easy to use and will help you. Open up Maple and go to Tools -> Tutors ->Linear Algebra -> Maple Inverses.

Test 1

The test will be on the first Tuesday slot of Week 5. That is Tuesday 25 February at 11:00 in B214. Ye have the test that I gave last year in the notes.

Week 3

We solved some 3×3 linear systems using the techniques of Week 2. We spoke about the determinant of a matrix and covered Cramer’s Rule. This concluded Chapter One.

Week 4

In Week 4 we will begin our study of Chapter 2, Vector Algebra. We will initially take a dual algebraic/geometric viewpoint.

Tutorials

If you have the 09:00 Lab on Fridays your tutorial will be
  • Friday 11:00 in B260
Otherwise your tutorial will be
  • Friday 13:00 in B245

Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

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 answer which explains the significance of the determinant.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MS3011: Week 6

February 13, 2014 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.

Week 6

We finished our study of the Logistic Family by analysing when zero was attracting/repelling and when \displaystyle \frac{\mu-1}{\mu} was attracting/repelling. We summarised our results in a bifurcation diagram

bifurcation

The green line corresponds to attracting fixed points and the red, dashed line to repelling fixed points. The graph is of fixed points vs \mu.

The fact that there are no attracting fixed points for \mu>3 indicates that the behaviour is more complicated when the growth rate, \mu, gets large. We could have periodic behaviour and perhaps more strange, chaotic behaviour.

We studied therefore the case where \mu=4. We said that for \mu=4, Q_\mu is symmetric about x=1/2 and is unimodal. We showed that  Q_4^n has 2^{n-1} branches and hence 2^n period-n points.

Week 7

In Week 7 we will finish of our study of the Tent Mapping and perhaps begin our study of the Doubling Mapping.

Exercises

I have emailed ye a copy of the exercises and ye have looked at questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye were ready for Wednesday’s Test). For the Week 7 tutorial you should look at these first and then backtrack and look at questions 30-37.

Test and Other CA

The results of the test will be revealed in time.

The Concept MCQ will take place on Wednesday 26 February Week 8 in WGB G05. It will be a half hour test and starts at 10.25 and runs until 10.55.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

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 finding a formula for the iterates of Q_1(x).

MATH7021: Week 2

February 7, 2014 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.

Manuals

Please get your manual ASAP from the Copy Centre. They are priced at E12.

Week 2

In Monday’s tutorial we looked at forward differences. In lectures we introduced Lagrange and Least Squares Interpolation.

Week 3

In Monday’s tutorial we will look at Least Squares. In lectures we will start the Chapter on Linear Algebra.

Test 1

Provisionally I am looking at Week 6. Ye will get a sample test (format only) and notice two weeks in advance.

Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

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 why we use Least Squares.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click on the action link button next to the course and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

MATH6040: Week 2

February 6, 2014 in MATH6040 | 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.

Week 2

We learned how to find the inverse of a 3×3 matrix using Gaussian Elimination and how to solve matrix equations. We saw that simultaneous equations can be written as a matrix equation and we learnt how to solve these.

Week 3

We will solve some 3×3 linear systems using the techniques of Week 2. We will talk about the determinant of a matrix and perhaps talk about Cramer’s Rule.

Test 1

Provisionally I am looking at Week 5. Ye will get a sample test (format only) and notice two weeks in advance.

Tutorials

If you have the 09:00 Lab on Fridays your tutorial will be
  • Friday 11:00 in B260
Otherwise your tutorial will be
  • Friday 13:00 in B245

Exam Format

Four questions, do all as in the Winter 2013 paper which can be found by searching through Mathematics here.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

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 which goes into some of the background theory of what we are doing.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click on the action link button next to the course and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 2

February 6, 2014 in MATH6038 | 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.

Manuals

Please bring E10.50 with you next week for your manual if you haven’t done so already. Those of you who didn’t get one last week will next week.

Week 2

We looked at the solution space of linear systems.

Week 3

We will introduce matrices as linear maps. In Maple, we will learn how Maple handles linear systems with no and infinite solutions. We will also use the Maple Tutor to help us do the below exercises.

Independent Learning: Exerices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do — on P.20 — at this point include (Q.2 should not be there):

  • Q. 1
  • Q. 4
  • Q. 5-7
  • Q. 8
  • Q. 9

Question 3

From P.20… we start by putting the system into augmented matrix form:

\left[\begin{array}{ c c c | c } 1 & 1 & -1 & 1 \\ 1 & 2 & -2 & 0 \\ -2 & 1 & 1 & 1 \end{array}\right]

r_2\rightarrow r_2-r_1,\,r_3+2r_1,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 3 & -1 & 3 \end{array}\right]

r_3\rightarrow r_3-3r_2,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 2 & 6 \end{array}\right]

r_3\rightarrow \frac12 \cdot r_3,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & -1 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right]

r_2\rightarrow r_2+r_3,\,r_1\rightarrow r_1+r_3,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]

r_1\rightarrow r_1-r_1,

\displaystyle \left[\begin{array}{ c c c | c} 1 & 1 & 0 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]

\Rightarrow z=3,\,y=2\text{ and }x=2.

Maple Online & Wolfram Alpha

I am aware that some of us have not been able to install Maple… I have got to here:

Maple Download

If  you cannot get to this screen please email me and I might be able to help (sadly I don’t have admin rights on the computer I am working on right now so cannot find out do I need the 32 bit or the 64 bit…).

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Test

Provisionally I am looking at Week 7. Ye will get a sample test (format only) and notice two weeks in advance.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

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 the solution set of a linear system.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MS3011: Week 5

February 6, 2014 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.

Week 5

We began our study of the Logistic Family. We postulated the equation as a model of population growth with two assumptions and ended up with

Q_\mu(x)=\mu x(1-x)

where x\in[0,1] can be interpreted as the proportion of a maximum population with a growth rate \mu\in[0,4].  We began analysing when zero was attracting/repelling and when \displaystyle \frac{\mu-1}{\mu} was attracting/repelling. We were rudely interrupted by the fire drill!

 

http://kkcb.com/one-of-the-best-scenes-of-the-office-is-dwights-fire-drill/

Week 6

In Week 6 we will finish of our study of the Logistic Mapping and perhaps begin our study of the Tent Mapping.

Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye are ready for Wednesday’s Test).

Test and Other CA

The test will take place on February 12 in Week 6. 

Everything up to but not including section 3.4 in the typeset notes is examinable. We have everything for the test covered. Please find a sample test and the test I gave last year after page 31. in these typeset notes.

The following theorems from the notes are examinable: the very bottom of page 12 and start of page 13. Also the Fixed-Point Factor-Theorem (which was called such on the board. It is not in the notes but is found here. When I say examinable you should be able to

  • state the theorem
  • prove the theorem
  • understand the theorem and the proof

Learning off the proof letter by letter won’t do you!

The Concept MCQ will take place in Week 8.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

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 finding a formula for the iterates of Q_1(x).

MATH6040: Week 1

February 3, 2014 in MATH6040 | 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.

Manuals

Please get your manual ASAP from the Copy Centre. They are priced at E12.

Week 1

After introductions, we looked at matrices and we introduced the idea of a matrix inverse.

Week 2

We will learn how to find the inverse of a 3×3 matrix using Gaussian Elimination.

Test 1

Provisionally I am looking at Week 5. Ye will get a sample test (format only) and notice two weeks in advance.

Exam Format

Four questions, do all as in the Winter 2013 paper.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

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 which asks why do we define matrix multiplication like we do.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click on the action link button next to the course and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH7021: Week 1

January 30, 2014 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.

Manuals

Please get your manual ASAP from the Copy Centre. They are priced at E12.

Week 1

After introductions, we looked forward differences. In fact we have 15 marks in the bag already.

Week 2

In Monday’s tutorial we will look at forward differences. In lectures we will introduce Lagrange and Least Squares Interpolation.

Test 1

Provisionally I am looking at Week 6. Ye will get a sample test (format only) and notice two weeks in advance.

Exam Format

Five questions do four:

Q. 1 — a question on each chapter.

Q. 2 — questions on chapter 1

Q. 3 — questions on chapter 2

Q. 4 — questions on chapter 3

Q. 5 — questions on chapter 4

As in this sample.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable is quite generous. You will get best results if you come to the helpers there with specific questions.

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 interesting application of forward differences.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click on the action link button next to the course and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘glyn advanced modern engineering math’ you will see that the library have an E-Book resource.

MATH6038: Week 1

January 30, 2014 in MATH6038 | 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.

Manuals

Please bring E10.50 with you next week for your manual. Those of you who didn’t get one last week will next week.

Independent Learning

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click on the action link button next to the course and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, you can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Week 1

After introductions, we looked at systems of linear equations and introduced the idea of writing them in augmented matrix form and simplifying using Gaussian Elimination.

Week 2

We will discuss how to determine if the linear system has a unique, infinite or no solution.

Test

Provisionally I am looking at Week 7. Ye will get a sample test (format only) and notice two weeks in advance.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

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 visualising the solutions of linear systems.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MS3011: Week 4

January 30, 2014 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.

Week 4

On Monday we spoke about cobweb diagrams that are a graphical method of locating fixed points and determining whether they are attracting, repelling or indifferent. At this point you should look at Section 2.4 of these notes (NOT mine).

In Section 3.1, 3.2 & 3.3, the cobweb diagrams suggest that if |f'|<1 (f' aka the slope) at the fixed point that the point is attracting and if |f'|>1 the fixed point is repelling.

A normal person would have proved this as per page 25 of the notes but I wanted to show you the beautiful contraction mapping principle and do it that way… we did conclude, loosely but correctly,

Theorem The: Contraction Mapping Principle

If f:I\rightarrow I is a contraction on a closed interval I, then f has a unique, attracting fixed point in I.

I mainly wanted to show ye this for the nice pictures! There is one here that I got off the internet but I think our blobs were better:

fixedpoint

O.K.

 

In the end what I did was a hodge-podge of the (correct) contraction mapping principle we did on Monday with the proof in the notes of the Fixed Point Dynamics Theorem below which isn’t really satisfactory to me. For next year I will probably try and take this approach but as my proof had a few holes, I am just going to say that you need to know the following:

Theorem: Fixed Point Dynamics

Suppose that f:\mathbb{R}\rightarrow \mathbb{R} is an iterator function with a fixed point at x_f\in \mathbb{R}. If f is differentiable in an interval I containing x_f then

  • If |f'(x_f)|<1, then x_f is an attracting fixed point
  • If |f'(x_f)|>1, then x_f is a repelling fixed point
  • If |f'(x_f)|=1, then we can make no conclusion and we call x_f an indifferent or neutral fixed point

Week 5

In Week 5 we will begin our study of the Logistic Mapping — rabbits!

Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions for the Week 5 tutorial.

  • 19 — a good question for more understanding for test
  • 23-25, 27-29 — good basic practise for the test. Do 25 and you can maybe leave the others
  • 26 — more theoretical than other questions of that type. You need to understand parts (a) to (c). These are the three theorems whose proofs are examinable.

The rest of these are not going to be on the test but the exam. If you forget them now don’t forget to look at before the exam to boost understanding. The theory is the same as before

  • 30-32 — not examinable on the test but have occurred in Q.2 of the exam. Week 4 theory: same as as 23-25, 27-29
  • 33 — full analysis of the dynamical system required using theory of Week 4.
  • 34-34 — given the orbit. Find the iterator function and analyse using theory of Week 4.
  • 36-37 — Newton-Raphson method is a dynamical system. Analyse using Week 4 theory.

As there are a lot of questions it might make sense to allocate so much time and say do (A)s first, then (B)s then (C)s or whatever.

Test and Other CA

The test will take place on February 12 in Week 6. Everything up to but not including section 3.4 in the typeset notes is examinable: we will have this covered by Februaray 3 but probably January 29. I have emailed ye a copy of a sample test.

The Concept MCQ will still take place in Week 8.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

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 where the OP didn’t understand why roots of f(x)-x are roots of f^2(x)-x.