We had one lecture and after listening to me go on about the importance of mathematics to your programme we started the first chapter on Sets and Relations. We saw something new with the concept of the power set of a set.
In Week 2 we will look more at sets and set identities, and explore Cartesian Products and perhaps introduce relations.
Tutorials start properly in Week 2.
I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. If you are a little worried about your maths this semester, perhaps after the Quick Test (emails about the Quick Test will be sent on Monday) or in general, you need to be aware of this resource. As you can see the timetable is quite generous:
Maths/Statistics Support in the ALC (Academic Learning Centre)
Monday 
1.00pm2.00pm 
D259 

4.00pm5.00pm 
D259 


Tuesday 
1.00pm2.00pm 
D259 

4.00pm6.00pm 
D259 
Thursday 
10.00am12.00pm 
D259 
Friday 
11.00am 1.00pm 
D259 
You will get best results if you come to the helpers there with specific questions.
There will be a 15% InClassTest in Week 5/6. Expect notice two weeks in advance and a sample test about 10 days in advance.
Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>
The manuals are available in the Copy Centre. Please purchase ASAP. More information has been sent via email.
Tutorials, which are absolutely vital, start next week.
In week one we had one and a half classes. One class was given over to a general overview of MATH7019 and we spent about half an hour introducing the topic of Curve Fitting.
We will introduce Lagrange Interpolation and start talking about Least Squares curve fitting.
I would urge anyone having any problems with material that isn’t being addressed in the tutorials to use the Academic Learning Centre. If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, you need to be aware of this resource. As you can see the timetable is quite generous:
Maths/Statistics Support in the ALC (Academic Learning Centre)
Monday 
1.00pm2.00pm 
D259 

4.00pm5.00pm 
D259 


Tuesday 
1.00pm2.00pm 
D259 

4.00pm6.00pm 
D259 
Thursday 
10.00am12.00pm 
D259 
Friday 
11.00am 1.00pm 
D259 
You will get best results if you come to the helpers there with specific questions. Next week, some students will receive slips detailing areas of maths that they should brush up on.
Assessment 1 will have a handin date around Week 5. Definite date to follow. The Assignment is in the manual but I must also send on your personal data sets.
Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>
The manuals are available in the Copy Centre and must be purchased as soon as possible. More information has been sent out via email.
Tutorial for BioEng2A: Thursdays at 12:00 in B180 A272 STARTS THURSDAY 20/09
Tutorial for BioEng2B: Mondays at 17:00 in B189 STARTS MONDAY 24/09
Tutorial for SET2: Mondays at 9:00 in E15 STARTS MONDAY 24/09
If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Next week, some students will receive slips detailing areas of maths that they should brush up on. The timetable is as below:
Maths/Statistics Support in the ALC (Academic Learning Centre)
Monday 
1.00pm2.00pm 
D259 

4.00pm5.00pm 
D259 


Tuesday 
1.00pm2.00pm 
D259 

4.00pm6.00pm 
D259 
Thursday 
10.00am12.00pm 
D259 
Friday 
11.00am 1.00pm 
D259 
We only had one lecture but began our study of Chapter 2, Vector Algebra by studying the difference between a scalar (single number) and a vector (list of numbers).
We will look at how to both visualise vectors and describe them algebraically. We will learn how to find the magnitude and direction of a vector, add them and scalar multiply them. We will speak about displacement vectors and introduce the vector product known as the dot product.
The test will probably be the Monday of Week 5: if progress with the vectors material is slow, we may push this out to Week 6. Official notice will be given in Week 3 (or Week 4 if necessary). There is a sample test in the notes.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>Abstract Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.
]]>What is a line? In Euclidean Geometry we usually don’t define a line and instead call it a primitive object (the properties of lines are then determined by the axioms which refer to them). If instead points and line segments – defined by pairs of points – are taken as the primitive objects, the following might define lines:
Geometric Definition Candidate
A line, , is a set of points with the property that for each pair of points in the line, ,
.
In terms of a picture this just says that when you have a line, that if you take two points in the line (the language in comes from set theory), that the line segment is a subset of the line:
Why is this objectively not a good definition of a line.
Once we move into Cartesian\Coordinate Geometry we can perhaps do a similar trick. We can use line segments, and their lengths to define slope, (slope = rise over run) and then define a line as follows:
Algebraic Definition Candidate
A line, , is a set of points such that for all pairs of distinct points , the slope is a constant.
This means that if you take two pairs of distinct points in a line , and then calculate the slopes between them, you get the same answer, and therefore it makes sense to talk about the slope of a line, .
This definition, however, has exactly the same problem as the previous. The definition we use isn’t too important but I do want to use a definition that considers the line a set of points.
We can use such a definition to derive the equation of a line ‘formula’ for a line of slope containing a point .
Suppose first of all that we have an axis and a point in the line. What does it take for a second point to be in the line?
The point is on the line if and only if the slope between and is equal to :
.
For any line at all this can be rewritten as , with the slope and the coordinate when : i.e. the intercept:
Parallel and perpendicular are then relations on the set of lines . This gives yet another definition of a line (that does not, by the way, include vertical lines. Again, we can include them with small tweaks to this presentation).
Algebraic Definition
A line, , of slope and intercept , is the set of solutions (in the plane) to the equation
.
A point is on the line if and only if it satisfies the equation:
.
It is probably more natural to come at these concepts first from a geometric angle (pun not intended). There are various definitions. For example, The Penguin Dictionary of Mathematics gives the following description:
Describing lines, curves, planes, or surfaces that are always equidistant, and that will never meet no matter how far they are produced. Parallel lines and curves must both lie in the same plane.
Whatever the definition, we know a pair of parallel lines when we see them.
One implication of this definition is that a line is not parallel to itself. This illustrates a feature of mathematics that is perhaps not appreciated at the primary and secondary levels, namely that
It’s sometimes hard for people learning mathematics, who naturally feel that mathematics is an objective discipline, to hear that many things are actually a matter of convention.
We have seen a similar thing above: we usually take lines themselves as primitive objects and define line segments in terms of lines. It seems to be possible to define lines in terms of line segments (although the two definitions above do not achieve this).
There is a choice here and the choice made dictates the detail of the sequel but not the essence… it is going to be slightly easier for me to present things if I say that a line is not parallel to itself (although if we do say this we get that ‘parallel’ is an equivalence relation). There are other choices but here is the definition that I will use is as follows.
Geometric DefinitionA line is parallel to a line , written , if , the empty set.
So rather than explicitly including the equidistance I am just going to say that a pair of lines are parallel if they do not intersect. I am almost completely sure that equidistance is a consequence of the above definition (perhaps with an additional axiom from Euclidean Geometry).
What we will show now is that parallel lines have the same slope.
Suppose that and are two parallel lines with equations:
and .
By the Geometric Definition, these lines do not intersect. That means there is no point that satisfies both equations, so no simultaneous solution to the equations. This means there is no solution to:
(*)
.
This clearly is a solution unless there is a division by zero, that is unless .
That is, parallel lines have the same slope. Note that (*) has a solution if but this is not allowed by the definition of parallel that I am using.
This suggests the following algebraic definition for parallel.
Algebraic Definition
A line is parallel to a different line , written ,if the lines have the same slope.
If we think of slope as measuring steepness/direction, this definition is quite natural.
If we take this as the definition then we can derive the geometric definition. Suppose that and are parallel. Therefore their equations are
and ,
with .
To find the intersection we solve the simultaneous equations. Doing this leads to which is false (the two lines must be different according to the definition). Therefore there is no intersection.
We have shown (with the restriction of lines to nonvertical lines) that
Geometric Definition Algebraic Definition, and
Algebraic Definition Geometric Definition
This implies that the definitions are equivalent. This post introduces the idea of Duality: in this case between Geometry and Algebra. The power of this duality is that sometimes it is easier to consider ideas/questions in the geometry picture, and sometimes it is easier to consider the algebra picture. The discussion here shows that when convenient we can think of parallel lines as being lines that do not intersect (geometry (or maybe even a set theory picture?)), and when convenient we can think of them as (distinct) lines that have the same slope; and that these are equivalent pictures.
Now we try the same kind of argument with perpendicularity.
Again, we should know perpendicular lines when we see them:
Lines are perpendicular when the angle between them is a right angle, . We don’t have to write down a good definition of angle: let us just say that an ordered triple defines an angle, with , the ‘common’ point:
is perpendicular to because .
As there are four angles involved, with two of them equal, and they add up to , we can define the angle between two lines as the smallest of the four and it is automatically less than or equal to .
This time, the candidate for an algebraic definition, using slopes, does not seem natural (although it is possible to use vectors and the dot product to generate a natural algebraic definition (that has the great property of being generalisable to more abstract settings)). Therefore we will simply write down the Geometric Definition and then prove an Algebraic Condition that holds for perpendicular lines. Again we restrict to nonvertical lines.
Geometric Definition
Lines and are perpendicular, written , if the angle between and is .
Now consider perpendicular lines and with slopes and :
Without affecting the slopes of and we can consider a new axis that has the same scale as the old but has as origin at :
So we can assume if we want that the perpendicular lines both go through the origin. Now go to and drop perpendiculars up and down to and as shown:
Now if we call the point by , note that and are both rightangled triangles and so Pythagoras Theorem applies:
and .
Calculate the length of using slope.
.
Similarly .
Now consider the rightangled triangle with hypotenuse and other sides such that and .
Apply Pythagoras Theorem to to show that:
Algebraic Condition
Lines and are perpendicular, written , if .
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If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.
On Monday we finished the module by looking at triple integrals.
The Wednesday 09:00 lecture will be a tutorial. In this class and your usual tutorial we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.
As next Monday is a bank holiday, we will begin the Summer 2017 Paper (in your notes) revision on Thursday.
In the Wednesday 09:00 lecture we will continue working on the Summer 2017 Paper and hopefully finish it before the end of the Thursday 10:00 lecture.
If we finish the Summer 2017 paper early, any extra time (probably just Thursday but maybe Wednesday if we go fast) will be dedicated to onetoone help.
Wednesday’s tutorial will go ahead as normal with onetoone help.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..
]]>
If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.
Assignment 2 has been corrected and your results emailed to you.
We spent three lectures looking at double integrals, in particular their application to second moments of area. We set of integration over a cylinder by looking at polar coordinates.
In the Wednesday tutorial we worked on the p. 163 (primarily) and p. 182 exercises.
On Monday we will finish the module by looking at triple integrals.
We will therefore have three tutorials where we will look at the P.182, P. 192, P. 163, & P.116 exercises. If these are completed you will be recommended to revise either by trying Chapter 1 & 2 exercises or perhaps by looking at the Summer 2017 paper.
We will go through last year’s exam on the board and then I will answer your questions if there are any. If there are none I will help onetoone.
Please feel free to ask me questions about the exercises via email or even better on this webpage.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..
]]>
If you would like to submit anonymous feedback on this module/lecturer, you may do so here. This link will be open until Friday May 11 2018.
We looked at the normal distribution.
In Maple looked at Binomial and Poisson random variables.
The Maple Test should take no more than one hour but I am giving ye extra time. For various reasons, I have decided to schedule next week’s class as:
The Maple Test will be open book. You have a sample Maple Test (this is also in the notes) with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2 (Lab 4).
We will speak about sampling in more detail and also introduce control charts.
We will hold a review class on Wednesday 9 May in the usual room. First off, the layout of your exam is the same as Autumn 2016 (in the back of your notes): do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.
I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.
If you have missed a lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.
Go through the missed lab on your own, doing all the exercises in Maple. Save the worksheet and email it to me.
The deadline for Maple Catch up is Friday May 11 2018.
Questions you can do include:
Use only determinants to determine if the following homogeneous system of linear equations has nonzero solutions:
I am not suggesting you should do all of these. It is recommended by the module descriptor that you do two hours of independent and directed learning every week but of course this isn’t feasible for everyone.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>We did a lot of probability — reliability block diagrams, the binomial distribution, the Poisson distribution.
Those who missed the class I recorded some of it here.
Ironically I recorded the same lectures in 2016 as you can see here.
We will look at the normal distribution and talk about sampling.
In Maple we will look at Binomial and Poisson random variables.
We will speak about sampling in more detail and also introduce control charts.
The Maple Test will be open book. You have a sample Maple Test with solutions (*the first with(Statistics) should be with(LinearAlgebra)). The Maple Test will not include anything from Chapter 2.
We will hold a review class on Wednesday 9 May in the usual room. First off, the layout of your exam is the same as Autumn 2016 (in the back of your notes): do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.
I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.
If you have missed a lab you have two options: either download Maple onto your own machine (instructions may be found here) or come into CIT at another time to use Maple.
Go through the missed lab on your own, doing all the exercises in Maple. Save the worksheet and email it to me.
Questions you can do include:
Use only determinants to determine if the following homogeneous system of linear equations has nonzero solutions:
I am not suggesting you should do all of these. It is recommended by the module descriptor that you do two hours of independent and directed learning every week but of course this isn’t feasible for everyone.
Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.
]]>