MATH6040: Spring 2019, Week 6

March 6, 2019 in MATH6040 | Leave a comment

Tuesday 19 March 2019, Week 8

As mentioned in previous weeks, I need to postpone the lecture of 19 March, Week 8.

This will now take place the next night, Wednesday 20 March 2019.

Two students have indicated that they cannot attend this class: I will record as much of the class as possible but my camcorder usually doesn’t have the battery nor memory to record all 2.5 hours… but I’ll do my best.

Outlook

As mentioned briefly in Week 6, for most students, Chapters 1 and 2 are easier and you will want to do well on them. Things are going to get a little harder for the rest of the semester and you will want to try and do homework (see below) regularly.

Week 6

We finished Chapter 2 by looking at Cramer’s Rule and then we did a Concept MCQ followed by tutorial time.

A video of a Cramer’s Rule Example

Week 7

We will do a quick revision of differentiation. We will then look at Parametric Differentiation and Related Rates.

If you want to look ahead here are two videos:

Homework Exercises

If you do any of the suggested exercises you can give them to me for correction. Please feel free to ask me questions about the exercises via email or even better on this webpage. Here are good exercises for Matrices. Feel free to try questions in the exercises that are not listed here (p.66, p.73, p.84,  or are in exam papers, see below):

  • P.63, Q.1-4
  • p.79, Q.1-2
  • p.95, Q.3-5

Test 2

Test 2, worth 15% and based on Chapter 3, will probably take place Week 11, 9 April. It might have to be pushed out to after Easter if we don’t make good progress on Chapter 3.

Test 1 Results

…and marking scheme have been emailed to you.

CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH7016: Spring 2019, Week 6

March 5, 2019 in MATH7016 | Leave a comment

VBA Assessment 1

VBA Assessment 1 is taking place this week, Week 6.

Tuesday 14:20-16:00 will run 14:20-16:10

Friday 09:05-10:45 will run 09:05-10:55

More information in the Week 4 weekly summary.

In the Week 5 VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

Written Assessment 1

Written Assessment 1 takes place Tuesday 12 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up – and replaces Section 1.6.1 of the manual.

However there are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.64). Some examples of questions I could ask include:

ODEs in Engineering

p.13, Examples 1-4; p.15, Q.1-4

General Theory

Example, p. 15; p.34 Example

Maclaurin/Taylor Series

Examples 1 & 2 on p. 24; Q. 1 on p.27

Euler Method

p.29, Examples 1-4; p. 38, Q.1-5, 8-9

Three Term Taylor Method

p. 35, Examples 1-2; p.39, Q.7, 10-14

Heun’s Method

p.38, Q. 6; p. 42, Examples 1-2l p. 47, Q.4-5

Second Order Differential Equations

p.50, Example. p.51, Example. p.55, Q. 1-3, 5-14

Read the rest of this entry »

Difficult Connected Particles Question

February 27, 2019 in Grinds, Leaving Cert Applied Maths | Leave a comment

When I say difficult, I mean difficult in comparison to the usual standard of Higher Level Leaving Cert Applied Maths Connected Particles Questions

Question (a)

Consider the following:

part1

A rectangular block moves across a stationary horizontal surface with acceleration g/5 (the question had g/5 m/s{}^2 but the m/s{}^2 is repeated as g=9.8 m/s{}^2 and so includes the unit).

There is a serious problem with this question and that is that the asymmetry in the problem means that there is an ambiguity: is the block moving left to right or right to left? I am going to assume the block moves from left to right. One would hope not to see such ambiguity in an official exam paper.

Two particles of mass M placed on the block, are connected by a taut inextensible string. A second string passes over a light, smooth, fixed pulley to a third particle of mass 2M which presses against the block as shown in the diagram.

(i)

If contact between the particles and the block is smooth, find the magnitude and direction of the resultant forces acting on the particles.

Solution

Note firstly that there are two accelerations at play. The acceleration of the block relative to the horizontal surface, g/5, and the accelerations of the particles relative to the block, say a:

part2

We draw all the forces (I lazily didn’t add arrows to the force vectors):

part3

We know that the normal forces for the particles on top of the block because their vertical acceleration is zero and so the sum of the forces in that direction must be zero, and as the down forces are equal for both, necessarily the up forces must be equal too.

Read the rest of this entry »

MATH7021: Spring 2019, Week 5

February 27, 2019 in MATH7021 | Leave a comment

Assignment 1

Assignment 1 has a hand-in time and date of 12:00 Friday 1 March (Week 5). Submit in class or to A283.

Work that is handed in late will be assigned a mark of ZERO so hand in what you have one time.

More information in last week’s Weekly Summary.

One final warning: do not give your work to others to copy. If there is a lack of originality of presentation I will be dividing marks between those who copy each other and the person who did the original work will be penalised along with those who copy them.

Week 5

We finished our work on Chapter 2 — the method of Undetermined Coefficients — Wednesday PM. The Thursday class was (will be as I write) a tutorial.

Week 6

On Monday we will have a tutorial on Undetermined Coefficients.

On Wednesday AM we will have a Concept MCQ, and then crack into Chapter 3, by looking at “The Engineer’s Transform” — the Laplace Transform.

Study

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

Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH6040: Spring 2019, Week 5

February 27, 2019 in MATH6040 | Leave a comment

Tuesday 19 March 2019, Week 8

As mentioned in previous weeks, I need to postpone the lecture of 19 March, Week 8.

Three possibilities to catch up are:

  • the next night, Wednesday 20 March 2019
  • the Tuesday before Easter, Tuesday 16 April 2019
  • the Wednesday after Easter, Wednesday 24 April 2019

Please fill in this Doodle poll by selecting all the days that you can attend.

Hopefully we can find a day that suits everyone but if people cannot make a day that is otherwise popular I will record the lecture.

Test 1 – Results

I hope to have these out to you within 24 hours.

Test 2

Test 2, worth 15% and based on Chapter 3, will probably take place Week 11, 9 April (or possibly Week 12, 30 April if we don’t go for a class Wednesday 20 March).

Week 5

We had our test and then we talked about linear systems, and determinants.

We had no tutorial time.

Week 6

We will finish Chapter 2 by looking at Cramer’s Rule. When we finish talking about Cramer’s Rule we will do a quick revision of differentiation, hopefully including some tutorial time.

CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH7016: Spring 2019, Week 5

February 26, 2019 in MATH7016 | Leave a comment

VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (5 & 8 March) in your usual lab time.

Tuesday 10:05-11:45 will run 10:05 to 11:55

Tuesday 14:20-16:00 will run 14:20-16:10

Friday 09:05-10:45 will run 09:05-10:55

More information in last week’s weekly summary.

In the Week 5 VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

Written Assessment 1

Written Assessment 1 takes place Tuesday 12 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up – and replaces Section 1.6.1 of the manual.

However there are far more things I could examine.

Roughly, everything up to but not including Runge Kutta Methods (p.64). Some examples of questions I could ask include:

ODEs in Engineering

p.13, Examples 1-4; p.15, Q.1-4

General Theory

Example, p. 15; p.34 Example

Maclaurin/Taylor Series

Examples 1 & 2 on p. 24; Q. 1 on p.27

Euler Method

p.29, Examples 1-4; p. 38, Q.1-5, 8-9

Three Term Taylor Method

p. 35, Examples 1-2; p.39, Q.7, 10-14

Heun’s Method

p.38, Q. 6; p. 42, Examples 1-2l p. 47, Q.4-5

Second Order Differential Equations

p.50, Example. p.51, Example. p.55, Q. 1-3, 5-14

Read the rest of this entry »

Publication

February 25, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups, Journal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

Abstract

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group \widehat{S_n} as well as on the truly quantum groups of Sekine.

MATH7021: Spring 2019, Week 4

February 21, 2019 in MATH7021 | 1 comment

Assignment 1

Assignment 1 has a hand-in time and date of 12:00 Friday 1 March (Week 5). Submit in class or to A283.

Read the P.51 and P.52 instructions carefully. You will be submitting an Excel file, and written work, including a print out of your Excel work.

Note in particular:

  • Work submitted after the deadline will be assigned a mark of ZERO. Hand up whatever you have on time.
  • Only Partial Pivoting has to be done using Excel.
  • Note that if you are doing Gaussian Elimination by hand you must use exact fractions and square roots rather a decimal approximation.
  • I advise that you do the questions out roughly first because small mistakes are inevitable.

The files you need to complete this assignment have been emailed to you. If you don’t want to calculate your c_i and P they are calculated in MATH7021A1 – Student Data.

We have now covered enough in class for you to do all of the Assignment. I recommend that you start ASAP.

WARNING!

This gives a good opportunity for collaboration but remember collaboration does not mean one student solving the problem and everyone else copying that student’s work. I demand originality of presentation here and you should at least understand what you hand up. If you are unsure of what I mean by this please email me immediately as if I have students who have clearly copied the answer word-for-word from another student they will all be sharing the marks.

Start early so you have enough time to complete the assignment properly and get good learning from it.

THIS IS A LEARNING ACTIVITY NOT JUST A GRADED ACTIVITY. THE CHAPTER ONE EXAM QUESTION IS WORTH 24.5% OF YOUR FINAL GRADE WHILE THIS ASSIGNMENT IS WORTH JUST 15%. THINK ABOUT WHAT THIS MEANS.

Regarding Q. 1.3.5, Assignment 1, on P.62 of the manual. The intention with Q. 1.3.5 (b) really is for you to engage in some problem solving skills to come up with a clever way of implementing the Jacobi Method in Excel.
It should still be doable by hand but if it takes a large number of iterations to converge (to two significant figures), Excel is far more suitable.
It is possible that it could take a small number of iterations to converge to two significant figures (say two or three iterations) — which is no problem by hand — but potentially it could take more (at least six). I don’t really want people spending loads of time doing iterations by hand, so I will give 3/4 marks for part (b) if you do six iterations by hand. If you want to keep going – by hand – until convergence (to two significant figures) you can of course get the 4/4 marks – but you need to ask yourself is it worth your time to keep going for the sake of one mark (out of 60… out of 15% —- that is 0.25% of your final grade).
If it converges with fewer than six iterations then happy days for you, you can get 4/4.
If it doesn’t, you might be better off trying to come up with a way of doing the question in Excel if you really want all the marks.
You can still answer part (c) if you do six iterations and do not yet have convergence.

Read the rest of this entry »

MATH6040: Spring 2019, Week 4

February 20, 2019 in MATH6040 | 2 comments

Test 1

Test 1, worth 15%, takes place from 19:00 to 20:05 sharp, Tuesday 26 February in the usual lecture venue. There is a sample on P.45 of the notes to give you an idea of the length and layout only.

Almost everything in Chapter 1 is examinable. This means:

  • P.23, Q.1-10
  • P.32, Q.1-8 [Q.9 is too long and Q.10 is not examinable]
  • P.39, Q.1-11

Additional practise questions may be found by looking at past exam papers (usually vectors are Q. 1, sometimes Q. 2).

You will want to be familiar with all the concepts in the Vector Summary, P. 41-44.

If you want questions answered you have two options:

  • ask me questions via email, perhaps with a photo to show your work
  • ask me questions via the comment function on this website

Week 4

We had Concept MCQ about vectors and then we started looking at Chapter 2: Matrices. We did some examples of matrix arithmetic and looked at Matrix Inverses — “dividing” for Matrices. This will allow us to solve matrix equations. Here find a note that answers the question: why do we multiply matrices like we do?

Read the rest of this entry »

MATH7016: Spring 2019, Week 4

February 19, 2019 in MATH7016 | 1 comment

VBA Assessment 1

VBA Assessment 1 will take place in Week 6 (5 & 8 March) in your usual lab time. You will not be allowed any resources – but the library of code (p.148) and these formulae will appear on the assessment:

relevant

The following is the proposed layout of the assessment:

Q. 1: Numerical Solution of Initial Value Problem [80%]

Examples of initial value problems that might be arise include:

  • Damping

\displaystyle \frac{dv}{dt}=-\frac{\lambda}{m}v(t);           v(0)=u

  • The motion of a free-falling body subject to quadratic drag:

\displaystyle \frac{dv}{dt}=g-\frac{c}{m}v(t)^2;           v(0)=u

  • Newton Cooling

\displaystyle \frac{d\theta}{dt}=-k\cdot (\theta(t)-\theta_R);           \theta(0)=\theta_0

  • The charge on a capacitor

\displaystyle \frac{dq}{dt}=\frac{E}{R}-\frac{1}{RC}q(t);           q(0)=0

Students have a choice of how to answer this problem:

  • The full, 80 Marks are going for a VBA Heun’s Method implementation (like Lab 3).
  • An Euler Method implementation (like Lab 2), gets a maximum of 60 Marks.

You will be asked to write a program that takes as input all the problem parameters, perhaps some initial conditions, a step-size, and a final time, and implements Heun’s Method (or possibly Euler’s Method): similar to Exercise 1 on p. 122 (except possibly implementing Heun’s Method) and also Exercise 1 on p.128 (except without the “conditional” derivative).

If you can write programs for each of the four initial value problems above you will be in absolutely great shape for this assessment.

Q. 2: Using your Program [20%]

You will then be asked to use your program to answer a number of questions about your model. For example, assuming Heun’s Method is used, consider the initial value problem (3.7) on p. 119.

  1. Given, v_0=0.2, m=3, \lambda=1.5, h=0.01, approximate v(0.3).
  2. Given, v_0=0.4, m=30, \lambda=1.5, h=0.1, investigate the behaviour of v(t) for large t.
  3. Given v_0=0.2, m=0.1, \lambda=1.5, h=0.5, T=10, run the Heun program. Comment on the behaviour of v(t). Run the same program except with h=0.05. Comment on the behaviour of v(t).
  4. Given, v_0=0, m=3, \lambda=1.5, h=0.1, T=2, run the Heun program. Comment on the behaviour of v(t).

Week 4

We finished off a Three Term Taylor Method example and spoke again about Heun’s Method.

We also introduced second order differential equations and saw how to attack them numerically. In particular we looked at a real pendulum.

In VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

Week 5

In the morning class we will finish looking at second order differential equations.

In the afternoon we will begin a quick study of Runge-Kutta Methods.

In VBA we have MCQ III and look at Lab 4, on Second Order Differential Equations.

Local vs Global Error

LocalvGlobal

Assessment

The following is a proposed assessment schedule:

  1. Week 6, 20% First VBA Assessment, Based (roughly) on Weeks 1-4
  2. Week 7, 20 % In-Class Written Test, Based (roughly) on Weeks 1-5
  3. Week 11, 20% Second VBA Assessment, Based (roughly) on Weeks 6-9
  4. Week 12, 40% Written Assessment(s), Based on Weeks 1-11

Study

Study should consist of

  • doing exercises from the notes
  • completing VBA exercises

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

Ungraded Concept MCQ League Table

To add a bit of interest to the Ungraded Concept MCQs, I will keep a league table.

Unless you are excelling, you are identified by the last five digits of your student number. AW is the number of attendance warnings received.

league2

Please ask questions in the lab about questions you have gotten wrong.