MATH7021: Week 5

February 28, 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

Please find a sample – (bit of a mistake here — change all the Ns to plus rather than minus).

Week 5

We continued finished our work on the Jacobi Method and the similar Gauss Siedel Method.  We summarised the five methods that we developed for solving linear systems; weighing up the pros and cons of each. Finally we started our work on Laplace Methods by talking about infinite integrals.

Week 6

Next week we will work slowly and deliberately developing our Laplace stuff. We will also have our test on Thursday.

Tutorials

This week coming it is, and I will be enforcing it

13:30 Groups S2 and S3 (A-L)

14:30 Groups S1 and S3 (M-Z)

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

tutorials

 

Diagonally Dominant Systems

We use the Jacobi & Gauss-Siedel Methods for solving diagonally dominant linear systems. I wasn’t sure where diagonally dominant systems show up in Civil Engineering but a quick Google shows me that they come up in finite element analysis. This is a technique to numerically solve differential equations that we can’t solve exactly: in some sense most useful differential equations can’t be solved exactly and finite element analysis is a method for approximating solutions.

MS3011: Week 8

February 28, 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.

Homework

Yes this homework is difficult. Take inspiration from JFK:

If you have problems interpreting the questions you can ask me in tutorials or on the this webpage. For dealing with your present confusion please be inspired by Mr. Feynmann:

Tutorial Venue

I have applied to get this changed to WGB G03… I didn’t get everything. The timetable is

  • This week coming 7 March LL2
  • 14 March WGB G03
  • 21 March WGB G03
  • 28 March WGB G03

Week 8

We finished our study of the tent mapping and began our study of the doubling mapping.

Week 9

We will finish off our work on the doubling mapping and then we will start talking about complex numbers; in particular the geometry of complex arithmetic and complex dynamics.

Complex dynamics is a truly beautiful area of maths that has lovely pictures like these:

mandel

Unfortunately we will only be scratching at the surface and won’t quite get this far!

Tutorials

Exercises for Thursday 7 March are to look at the following. Not a whole pile of new stuff covered so some revision.

Summer 2010 Question 3 (b)

Autumn 2010 Question 3(b)

Autumn 2009 Question 3

MATH6037: Week 5

February 28, 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

A test will be held in Week 7, 13 March 2013. Please find a sample attached (you will also be given these tables). 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

I understand that a few of ye have a very heavy schedule in Week 7. If and only if this is the case I am considering giving ye an opportunity to sit the test on Thursday evening 21 March at 19:10. Please email me if you fall into this bracket.

Revision

We will do some revision next week but really you should be trying exercises yourself. Please feel free to ask me questions via email or even better on this webpage.

Week 5

In Week 5 we looked at differentials and how they help in error analysis. We looked at the Bisection & Newton-Raphson Methods for approximating solutions of equations. We began looking at methods of approximate, numerical integration.

Week 6

In Week 6 we will do some revision, namely the sample test. I would strongly advise you to try the sample test before this class to see what you can do and what you can’t do. After this we will have a Maple session.

Week 7

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

MATH6040: Week 4

February 22, 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

The test will be held in Week 6 (Wednesday 6 March). Please find a sample. I have decided that the test will be on matrices only so everything in Chapter 1 is examinable. The test will be 50 minutes long. The sample is to give you an idea of the format and length.

Week 3

In Week 4 we will begin our study of Vector Algebra.

Week 4

In Week 4 we will continue our work on Vector Algebra including the Cross Product.

Tutorials

I have made a promise that I will make the matrix inverse easier to compute like the examples on p.22 and 23… this will be the case.

MATH7021: Week 4

February 21, 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.

Five students still owe me E10 for the notes. If this is a problem for you send me a confidential email. I don’t want to be chasing ye but if ye can’t get in touch with me I will have to.

Test 1

The test will be held in Week 6 (Thursday 7 March). Please find a sample – (bit of a mistake here — change all the Ns to plus rather than minus). The test will be on the material covered in weeks 1-4 inclusive so everything up to section 2.3 inclusive. The test will be 50 minutes long. The sample is to give you an idea of the format and length. However Q.1 will be on finite differences, Q. 2 will be on Least Squares, Q. 3 Gaussian Elimination and Q.4 Gaussian Elimination with Partial Pivoting. Lagrange Interpolation can sneak into Q. 1 or 2 while Cramer’s Rule can sneak into Q. 3 or 4.

Week 4

We continued our work on Gaussian elimination. We discussed Cramer‘s Rule; we explained why computers struggle with Gaussian elimination and learnt about Gaussian Elimination with Partial Pivoting. We started talking about the Jacobi Method: a way of quickly approximating solutions to linear systems: especially diagonally dominant ones.

Week 5

Next week we will develop the Jacobi Method and the similar Gauss Siedel Method. At this point we might stop and discuss the advantages and disadvantages of our five methods (Gaussian Elimination, Gaussian Elimination with Partial Pivoting, Cramer’s Rule, Jacobi Method, Gauss-Siedel Method). Depending on time we might begin talking about Laplace Methods…

Tutorials

Deadly serious: ye need to be A-standard on the first two chapters… they are doable for everyone with a bit of work. The later stuff will be challenging and will need a lot of work to reach a good standard.

This week 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 7

February 21, 2013 in MS 3011 | 6 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

I am afraid I am a little on the busy side and ye might be waiting until around March 8 for your results. Just to clarify, I had intended for ye to prove rather than show in question 1; however I wrote show. Therefore if you used Theorem 1 or 2 to answer you will have to get full marks. However at the same time some people understood show as prove. By and large these people are looking at almost 100% but if they drop marks elsewhere, I will give a bonus mark to students who proved in question 1 rather than showed.

Homework

Please find the Homework. Before you open it don’t be too alarmed: you only have to do ONE of the SIX options. All of the options are about dynamical systems & complex numbers in different areas of math:

  1. Discrete Mathematics, Number Theory & Abstract Algebra
  2. Probability
  3. Differential Calculus
  4. Integral Calculus
  5. Linear Algebra
  6. Complex Numbers

Therefore, if you are good at differential calculus, for example, you should have a look at option 3.

All of these questions are unseen to you and — with the exception of Q.6 — all require some knowledge of modules you are doing now or have done before. Although we have been concentrating on real-valued functions on the set of real numbers (i.e. f(x), etc.), a lot of the theory carries over into more general sets and functions, and this is the main learning outcome of this homework.

I am not going to pretend that this is an easy assignment, but I will say that clear and logical thinking will reveal that the solutions and answers aren’t ridiculously difficult: a keen understanding of the principles of dynamical systems and a good ability in one of the options should see you through.

The final date for submission is 12 April 2013 and you can hand up early if you want. You will be submitting to the big box at the School of Mathematical Science. If I were you I would aim to get it done and dusted early as this is creeping into your study time and is very close to the summer examinations.

Note that you are will be free to collaborate with each other and use references but this must be indicated on your hand-up in a declaration. Evidence of copying or plagiarism (although this is unlikely as these are original problems by and large) will result in divided marks or no marks respectively. You will not receive diminished marks for declared collaboration or referencing although I demand originality of presentation. If you have a problem interpreting any question feel free to approach me, comment on the webpage or email.

Ensure to put your name, student number, module code (MS 3011), and your declaration on your homework.

Tutorial Venue

I have applied to get this changed to WGB G03… I didn’t get everything. The timetable is

  • This week coming 28 February LL2
  • 7 March LL2
  • 14 March WGB G03
  • 21 March WGB G03
  • 28 March WGB G03

Week 6

We finished describing what a chaotic dynamical system is and began our study of the tent mapping

What next?

We won’t be long finishing off our work on the tent mapping and then we will commence our third special study: of the doubling mapping.

Tutorials

Exercises for Thursday 28 February are to look at the following. Not a whole pile of new stuff covered so some revision.

Autumn 2009 Q. 2(b), 4

Summer 2008, Q. 1(a), 2(a), (b), 3, 6

MATH6037: Week 4

February 21, 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.

Week 4

In Week 4 we looked at Multivariable Calculus and Partial Differentiation . We are going to use this stuff for error analysis. We did some Partial Fractions & Partial Differentiation in our Maple lab.

Next Week

We will begin to look at error analysis — very important for the analysis of physics experiments/data. We will hopefully begin some Numerical Methods.

Test

The test will be held in Week 7 (13 March). I will send ye a sample next Friday and the test we be on the material covered in weeks 1-5 inclusive. We will do some revision in Week 6 before Maple.

MATH6040: Week 3

February 15, 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.

Week 3

We learned how to use the matrix inverse to solve simultaneous equations. We also introduced Cramer’s Rule which is a method that is useful if you only want to find one of the variables rather than all of them. Cramer’s Rule uses determinants and we showed how to calculate the determinant of 2\times 2 and 3\times 3 matrices. We also explained that when the determinant of the coefficient matrix is zero, the Gauss-Jordan algorithm cannot work because the matrix is not invertible.

Week 4

In Week 4 we will begin our study of Vector Algebra.

Tutorials

If we can do the question on page 43 of the notes, then you are in a good place for matrix algebra. I have made a promise that I will make the matrix inverse easier to compute like the examples on p.22 and 23.

MATH7021: Week 3

February 15, 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.

Five students still owe me E10 for the notes. If this is a problem for you send me a confidential email.

Week 3

We started our work on linear algebra; specifically linear systems. We showed how to write a linear system in augmented matrix form and showed that the elementary row operations conserved the solutions. We loosed described the Gaussian elimination algorithm that would put the augmented matrix in reduced form. We said that solutions could have none, one or an infinite number of solutions and described how to tell from the reduced form. Depending on the scenario, the solutions are easily found via back-substitution.

Week 4

Next week we will continue to work on Gaussian elimination. We will discuss Cramer‘s Rule; we will see why computers struggle with Gaussian elimination and learn ways to get around this.

Tutorials

Deadly serious: ye need to be A-standard on the first two chapters… they are doable for everyone with a bit of work. The later stuff will be challenging and will need a lot of work to reach a good standard.

This week it is

13:30 Groups S2 and S3 (A-L)

14:30 Groups S1 and S3 (M-Z)

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

tutorials

Additional Notes

Read the rest of this entry »

MATH6037: Week 3

February 15, 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.

Week 3

In Week 3 we looked at Partial Fractions. They provide us with a way of integrating rational functions and will be very important later for when we study Laplace Methods.

Next Week

We will look start Multivariable Calculus and perhaps begin to look at its application to error analysis — very important for the analysis of physics experiments/data. We will do some Partial Fractions & Multivariable Calculus in our Maple lab.

Additional Notes

I skipped over some of the more complicated Partial Fractions stuff that we don’t need for exams. For completeness I have put it here.

Read the rest of this entry »