MATH6014: Sample Paper

March 8, 2011 in MATH6014 | Leave a comment

The first MATH6014 test will be held at 5 p.m. today. The test is worth 15% of your final mark. The test will be 50 minutes long and you must answer all questions. Question 3 is worth 40 marks; Questions 1 and 2 30 marks each. I will put the formula for the roots of a quadratic equation on the paper. Please find a sample here.

MATH7019: Week 5

March 4, 2011 in MATH7019 | 2 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 jippo@campus.ie and I will add you to the mailing list.

This Week

We developed our applications of step and impulse functions to beam equations. Relevant notes p.22 – 50

In the tutorial we worked on the sample test.

Next Week

We have a tutorial on Thursday. Please take this opportunity to nail down the differential equations. On the final exam there will be both a full question on beam equations and a full question on second order linear differential equations. Hence if you are comfortable with the material that is examinable in the test, feel free to move onto some of the exercises on beam equations.

Test Date

Tuesday 13 March at 6.45 p.m.

Timetable Changes

We are now going to schedule ourselves as follows:

Week 6: – & Tutorial

Week 7: Lecture/Test & –

Week 8: Lecture & Lecture

Week 9: Tutorial & Lecture

Week 10: Tutorial & Lecture

Week 11: – & Lecture

Week 12: Tutorial & Lecture

Sample Test Answers

Here I give you links to how I checked answers quickly using Wolfram Alpha. It takes Mathematica code (they are the same company) and will probably decipher your own stab at code also. Note that Wolfram Alpha gives us a lot more information than we need but that is the beauty of the thing really.

Question 1 — note there is a small typo here it should be 16y rather than just 16.

Question 2 — it’s not evaluating the constants using the boundary conditions for some reason… the answer is y(x)=-\frac{1}{9}e^{-8x}+\frac{1}{9}e^x

Question 3

Question 4 — the boundary conditions yield y(x)=-\frac{55}{2}e^{2x}+\frac{52}{3}e^{3x}+5x+\frac{25}{6}.

Question 5 — what is \theta here and how do we write (x-2)\theta(x-2)?

Question 6 — again we need to realise that the notation here is different to ours. Applying the boundary conditions we get y(x)=3[x-8]^3-3[x-2]^3+54x.

MATH6038: Week 5

March 4, 2011 in MATH6038 | 2 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 jippo@campus.ie and I will add you to the mailing list.

This Week

We looked at different types of data and how to present them. Notes here. When we look at histograms next week this section will be complete. In Maple we continued to boost our basic skills and looked at Maple’s LinearAlgebra package.

Next Week

I hope to finish looking at histograms. Then we will begin our study of probability, asking the question what is a random variable and when are random variables independent?

Reminder

The test will be held at 7.05 p.m. sharp on Wednesday 14 March.

Sample Test Answers

Here I give you links to how I checked answers quickly using Wolfram Alpha. It takes Mathematica code (they are the same company) and will probably decipher Maple code too (moreover Maple code with ‘adjustments’). Note that Wolfram Alpha gives us a lot more information than we need but that is the beauty of the thing really.

Question 1 (has a slight typo: should be -6z in the first equation)

Question 2

Question 3 — this implies that there are non-trivial solutions. Now that we know that it has non-trivial (non-zero solutions) so we can use Wolfram Alpha to solve for these (in terms of a parameter t.). Now let z=t so we have (x,y,z)=(t,t/2,t).

Bonus Questions

You could do without all the notation. The underlined would do.

1. Firstly the linear system can be written as a set of simultaneous equations:

a_{11}x_1+\cdots+a_{1n}x_n=b_1

a_{21}x_1+\cdots+a_{2n}x_n=b_2

\vdots+\cdots+\vdots=\vdots

a_{m1}x_1+a_{mn}x_n=b_m

The solution set is not changed by E_1 as this is simply writes the equations in a different order. Neither will E_2 change the solution (well if k\neq0) as multiplying one of the equations by a constant on both sides will not change the solution. Finally E_3 doesn’t change the solution set as we add the LHS_i (left-hand-side) of one equation to the LHS_j of another, while we add the constant b_i to b_j. However the LHS_i=b_i so we have just added the same thing to both sides and this does not change the solution.

2. A\mathbf{x}=\mathbf{b}\Rightarrow A^{-1}A\mathbf{x}=A^{-1}\mathbf{b}\Rightarrow I\mathbf{x}=A^{-1}\mathbf{b}\Rightarrow \mathbf{x}=A^{-1}\mathbf{b}.

3. You can solve the system for one rather than all variables.

Ideals and Positive Functionals: 3.4 The Gelfand-Naimark-Segal Representation

February 28, 2011 in C*-Algebras & Operator Theory, Research | Leave a comment

In this section we introduce the important GNS construction and prove that every C*-algebra can be regarded as a C*-subalgebra of B(H) for some Hilbert space H. It is partly due to this concrete realisation of the C*-algebras that their theory is so accessible in comparison with more general Banach spaces.

A representation of a C*-algebra A is a pair (H,\varphi) where H is a Hilbert space and \varphi:A\rightarrow B(H) is a *-homomorphism. We say (H,\varphi) is faithful if \varphi is injective.

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MA1008: Exercise 4.13

February 28, 2011 in MA 1008 | Leave a comment

The Question

The pressure, volume, and temperature of an ideal gas are related by the equation PV=8.31T (when pressure is measured in kilopascals). Find the rate at which the pressure is changing when the temperature is 300 K and increasing at a rate of 0.1 K s^{-1}, and the volume is 100 L and increasing at a rate of 0.2 L s^{-1}.

Solution

First of all, solving for P:

P(T,V)=\frac{8.31T}{V}.

Now we can go further and say that both T and V are functions of time, t. So we have:

P(T(t),V(t))=\frac{8.31T(t)}{V(t)}=8.31T(t)[V(t)]^{-1}.

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MATH6038: Week 4

February 26, 2011 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 jippo@campus.ie and I will add you to the mailing list.

This Week

We said that a homogeneous linear system was one where all the constants are zero. We then proved the following:

Proposition

Let A\mathbf{x}=\mathbf{0} is a homogenous linear system of n equations in n unknowns where A is the coefficient matrix and the variables are x_1,\,x_2,\,\dots,\,x_n ( i.e. \mathbf{x}=(x_1\,\,x_2\,\,\cdots\,\,x_n)^T. Then the following holds:

  1. If \det A=0 then the system has (an infinite number of) non-trivial solutions.
  2. If \det A\neq =0 then the system has (a unique solution given by) the trivial solution (x_1=x_2=\cdots=x_n=0).

MATH6038: Sample Test

February 26, 2011 in MATH6038 | Leave a comment

Consider this notice for the test on Wednesday 14 March at 7 p.m (Wednesday fortnight).

Please find a sample test here.

MATH7019: Week 4

February 24, 2011 in MATH7019 | 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 jippo@campus.ie and I will add you to the mailing list.

This Week

We introduced step and impulse functions to model loads on a beam and we wrote down the equations relating the loading, the shearing force, the bending moment and the deflection. Relevant notes p.23 to p.43.

In the tutorial we worked on the exercises on second order linear differential equations in these notes.

Next Week

We have a tutorial on Tuesday. In this we will try to shore up the second order differential equations and the second order differential equations involving step and impulse functions. Remember the integration of impulse and step functions is given by (where each of the arrows indicate an integration)

\delta(x-a)\rightarrow H(x-a)\rightarrow [x-a]\rightarrow\frac{[x-a]^2}{2}, and then

\int [x-a]^n\,dx=\frac{[x-a]^{n+1}}{n+1}.

Also each integration generates an addition constant C_1,\,C_2,\dots.

In next weeks lectures we will continue our work on the beam equations.

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MATH7019 Sample Test 1

February 22, 2011 in MATH7019 | 2 comments

Consider this notice for the test on Tuesday 13 March at 6.45 p.m (just under three weeks away) (note that there is still a small chance that this tell will be held on Thursday 8 March at 8.15 p.m.).

Please find a sample test here

Note that the format will be the same as this.

  1. Homogeneous Second Order Linear
  2. Homogeneous Second Order Linear with boundary/initial conditions
  3. Non-Homogeneous Second Order Linear
  4. Non-Homogeneous Second Order Linear with boundary/initial conditions
  5. Second Order Separable with Step and Impulse Functions
  6. Second Order Separable with Step and Impulse Functions with boundary/initial conditions

Q. 5 and 6 will be covered Thursday night. Note that for questions 1-4 the roots will not be complex — although they could be fractions or surds (i.e. ye might need the -b\pm\sqrt{...} formula).

Ideals and Positive Functionals: 3.3 Positive Linear Functionals

February 22, 2011 in C*-Algebras & Operator Theory, Research | 2 comments

For abelian C*-algebras we were able to completely determine the structure of the algebra in terms of the character space, that is, in terms of the one-dimensional representations. For the non-abelian case this is quite inadequate, and we have to look at representations of arbitrary dimension. There is a deep inter-relationship between the representations and the positive linear functionals of  a C*-algebra. Representations will be defined and some aspects of this inter-relationship investigated in the next section. In this section we establish the basis properties of positive linear functionals.

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