MS 2001: Week 6 (Test 1)

October 27, 2011 in MS 2001 | 1 comment

Test Results

First of all results are down the bottom. You are identified by the last four digits of your student number (or five if these four digits are shared by another student). The scores are itemized as you can see. At the bottom there are some average scores.

If you would like to see your paper or have it discussed please email me.

Students with no score were either absent in which case they score zero — or certified absent in which case their marks carry forward to the summer exam.

Solutions and Remarks

Solution are found here.

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Leaving Cert Project Maths: Proofs for 2012

October 25, 2011 in General, Grinds, Leaving Cert | 19 comments

With the new Project Maths programme being developed as we speak, diligent students might like to know which proofs are examinable under the new syllabus so they know which to look at.

It can be difficult to sift through the syllabi at projectmaths.ie but I have gone through them and here are the proofs required.

Hopf Algebras: Algebras

October 25, 2011 in Quantum Groups, Research | 1 comment

Taken from Hopf Algebras by Abe. I am not doing this over a commutative ring as he is doing but just over the field of complex numbers, \mathbb{C} — therefore some of my constructions will be simplified. Some of them might even be wrong.

In this section the notion of algebras will be introduced; and we will present some examples of such algebras. An algebra over \mathbb{C} is defined by giving a structure map.

Algebras

Suppose A=\{e_\lambda,0\}_{\lambda\in\Lambda} is a ring and we are given a ring morphism \eta_A:\mathbb{C}\rightarrow A. Then A can be viewed as a complex vector space. If we have

(ax)y=x(ay)=a(xy) for all a\in\mathbb{C} and x,\,y\in A

then A is said to be an algebra. Now the map defined by f(x,y)=xy turns out to be bilinear. Thus we obtain a morphism

\nabla_A:A\otimes A\rightarrow A.

From this fact, we see that an algebra A can be defined in the following manner. For a complex vector space A and morphisms \eta_A:\mathbb{C}\rightarrow A, \nabla_A:A\otimes A\rightarrow A, we have the following:

\nabla_A\circ(\nabla_A\otimes I_A)=\nabla_A\circ(I_A\otimes\nabla_A)

(the associative law)

\nabla_A\circ(\eta_A\otimes I_A)=\nabla_A\circ (I_A\otimes \eta_A)=I_A

(the unitary property)

Here, \nabla_A is said to be the multiplicative map of A, \eta_A the unit map of A, and together we call \nabla_A, \eta_A the structure maps of the algebra A.

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MS 2001: Week 5

October 19, 2011 in MS 2001 | Leave a comment

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

[EDIT] Definition Questions

If you really want to get good at your definition questions check out these (tough) exercises..

Next Week

We will have an additional tutorial instead of a lecture on Monday 24.

This Week

On Monday, we covered from (but not including) Definition 3.1.1 to (and including) Corollary 3.1.6. On Wednesday we went through Questions 1 & 2 from 2010/11’s Test 1 A and we did Q. 3 from the sample test.

In the tutorial we answered exercises Q. 1 (vii), 3 (b), 4(ii) & 8(iii) from Exercise Sheet 2.

Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Q. 1, 2, 3(i) & (iii), 4, 8(a) & (b) from Exercise Sheet 3.

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Hopf Algebras: Bialgebras and Hopf Algebras

October 14, 2011 in Quantum Groups, Research | Leave a comment

Taken from Hopf Algebras by Abe. This is not even nearly finished however I pressed publish instead of save draft… oh well. 

In this section, we give the definition of Hopf algebras and present some examples. We begin by defining coalgebras, which are in a dual relationship with algebras., then bialgebras and Hopf algebras as algebraic systems in which the structures of algebras and coalgebras are interrelated by certain laws.

1.1 Coalgebras

We define a coalgebra dually to an algebra. Given an algebra A and algebra homomorphisms  \Delta:A\rightarrow A\otimes A and \varepsilon:A\rightarrow\mathbb{C}, we call (A,\Delta,\varepsilon) or just A a coalgebra when we have:

(\Delta\otimes I_A)\circ \Delta=(I_A\otimes\Delta)\circ\Delta,

(the coassociative law).

(\varepsilon\otimes I_A)\circ\Delta=(I_A\otimes\varepsilon)\circ\Delta,

(the counitary property)

The maps \Delta and \varepsilon are called the comultiplication map and the counit map of A, and together they are said to be the structure maps of the coalgebra A.

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About Leaving Cert Applied Maths

October 12, 2011 in Leaving Cert Applied Maths | 4 comments

Applied maths could be defined as the use of mathematics in studying natural phenomena. The branch of applied maths studied at Leaving Cert level is Newtonian mechanics. Mechanics is the study of systems under the action of forces. Newtonian mechanics is concerned with systems that can be adequately described by Newton’s Laws of Motion. Systems that aren’t adequately described by Newtonian mechanics include systems with speeds approaching the speed of light, systems of extremely small particles and systems with a large number of particles. Hence Leaving Cert Applied Maths is the study of simple macro-systems that have moderate speeds.

Applied maths is essentially a further study of the mathematics of chapters 6 through 13 in Real World Physics; following which more specific and involved questions than those of Leaving Cert Physics may be posed and answered. The emphasis in applied maths is more on problem-solving than anything and reflecting this, the need for rote-learning is almost non-existent. The course content itself could be presented on one A4 sheet. The skills required for the course include:
• Capacity for interpretation and visualisation
• Ability for strategic problem solving
• Competency in mathematics

Anyone who is strong in higher level maths (A or B standard) and is self-motivated can achieve great success in applied maths – especially in conjunction with LC Physics. There is then a three for the price of two and a half effect as applied maths will help your physics, physics will help your applied maths and maths will help your applied maths. For those looking at the bigger picture, if you are intending on pursuing any science or engineering course in college, having done LC Applied Maths will really stand to you in your daunting first year — most 1st year physics and applied maths modules broadly cover the material of LC Applied Maths.

The statistics over the last number of years are that typically over 90% of students taking applied maths do higher level, and of these about a quarter achieve an A, over half achieve an A or B and nearly two-thirds achieve an honour. 90% of students pass. So with strong maths and good, effective application to the task at hand, good grades are there for the taking.

Some LC Applied Math Notes and some Solutions to selected problems. Section 6.2 deals with the Fundamental Theorem of Calculus and Section 6.4 has a little integration. See Chapter 10 of these LC Maths Lecture Notes for some more on integration.

MS 2001: Week 4

October 12, 2011 in MS 2001 | Leave a comment

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

Test 1

Firstly, some good news. I am going at a different pace to last year so in fact we have covered everything we need to do for Test 1 already. All of Chapter 1 and Sections 2.1, 2.2 and 2.3 of Chapter 2 are all that will be examinable in Test 1. This means that Section 2.4 Continuity on Closed Intervals and Chapter 4 Differentiability are not  examinable.

Please find the Sample, Test 1 A and Test 1 B (second two with solutions).

Question 1 will be taken from the exercise sheets, Question 2 from a past exam paper, and Question 3 will be on definitions. Q.1 is worth 4/12.5 or 32%, Q. 2 is worth 5/12.5 or 40% and Q. 3 is worth 3.5/12.5 or 28% (1 correct = 1 mark, 2 correct = 2 marks, 3 correct = 3 marks + 0.5 mark bonus).

For Q. 3 of the test, you need to know the following definitions: even, odd, increasing, decreasing, quadratic, roots, polynomial, rational function, absolute value, limit, one-sided limit, continuous at a point, continuous, composition. Q.3 is a harder question and the thinking behind this is that you can get 72% a bare first if you get all of Q.1 and Q.2 — but you will have to be even better than this to get a higher mark.

 

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LC Project Maths: Pensions

October 12, 2011 in Grinds, Leaving Cert | 3 comments

This is a short note covering pensions as done in LC HL Project Maths. I do not know how pensions are calculated “in the real world”.

Fixed Number of Payouts

Suppose you want a pension that will pay you €20,000 per year for 25 years after retirement. How much should you have in your pension fund on retirement in order to have this? Suppose further that money can be invested at 3% per annum.

Method 1

Suppose we need the pension fund to contain €X on retirement. Let P(t) be the amount of money in the pension fund after t years and suppose the pension fund is invested at 3%. Well, in the first year we need:

p(0)=X,

but then take out €20,000:

p(1)=X-20,000

We then accrue interest on this (capital + interest = (1+i)) — but then withdraw €20,000 at the end of the first year so:

p(2)=X(1+i)-20,000(1+i)-20,000.

Now this accrues interest but €20,000 is withdrawn:

p(3)=X(1+i)^2-20,000(1+i)^2-20,000(1+i)-20,000

\vdots

p(25)=X(1+i)^{24}-20,000[(1+i)^{24}+(1+i)^{23}+\cdots+(1+i)+1].

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Representations of C*-Algebras: Irreducible Representations and Pure States

October 5, 2011 in C*-Algebras & Operator Theory, Research | 1 comment

Taken from C*-Algebras and Operator Theory by Gerald Murphy.

This section is concerned with positive linear functionals and representations. Pure states are introduced and shown to be the extreme points of a certain convex set, and their existence is deduced from the Krein-Milman theorem. From this the existence of  irreducible representations is proved by establishing a correspondence between them and the pure states.

If (H,\varphi) is a representation of a C*-algebra A, we say x\in H is a cyclic vector for (H,\varphi) if x is cyclic for the C*-algebra \varphi(A) (This means that cyclic vector is a vector x\in H such that the closure of the linear span of \{\varphi(a)x\,:\,a\in A\} equals H). If (H,\varphi) admits a cyclic vector, then we say that it is a cyclic representation.

We now return to the GNS construction associated to a state to show that the representations involved are cyclic.

Theorem 5.1.1

Let A be a C*-algebra and \rho\in S(A). Then there is a unique vector x_\rho\in H_\rho\in H such that

\rho(a)=\langle a+N_\rho,x_\rho\rangle, for a\in A.

Moreover, x_\rho is a unit cyclic vector for (H_\rho,\varphi_\rho) and

\varphi_\rho(a)x_\rho=a+N_\rho, for a\in A.

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MS 2001: Test 1

October 5, 2011 in MS 2001 | Leave a comment

The first in-class test will take place on 26 October 2011. Any material presented in class, up to and including 19 October is examinable. The test is worth 12.5% of your continuous assesment mark for MS 2001. A sample test (without solutions) — as well as last years test (with solutions) shall be posted here on 12 October 2011. Although if you are willing to look through the “MS 2001: Continuous Assessment” category you will find these easily enough.