Research Update 7

November 29, 2010 in Research, Research Updates | Leave a comment

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I have finished off section 1.4 including Atkinson’s Theorem and a first look at the unilateral shift. I have done exercises 1-7. In terms of progress, I am on p.31 of 265, with 13 exercises left in this section. Following discussions with my supervisor, I may be able to leave out sections 3.2, 3.5, 4.4, 5.2-6 and the whole of chapter 7.

MS 2001: Week 10

November 25, 2010 in MS 2001 | Leave a comment

Firstly; there will be no MS 2001 lecture on Monday 6 December at 3 p.m. Instead you will have an MS 2003 lecture at this time in WG G 08. The 12 p.m. MS 2003 lecture on Wednesday December 1 in WG G 08 will now be an MS 2001 lecture.   Indeed it will be the final MS 2001 lecture as Wednesday 8 December is a test day and the week after is review week. The morning lecture at 9 a.m. on Wednesday 1 December will still go ahead.

On Monday we wrote down the Second Derivative Test and the First Derivative Test. We showed that the First Derivative Test is superior as it can correctly handle all of the functions that the Second Derivative Test can and more (functions with vanishing second derivative and also functions that have points that are not differentiable).
On Tuesday we did Q. 1 & 2 from the sample. It was clear the sample test is too long and I will ensure that the actual test (Wednesday 8 December) isn’t as long.
On Wednesday we defined what it means for the graph of a function to be concave up or concave down. We defined a point of inflection to be a point on the graph of a function where the concavity changes. We then said that we had a lot of tools that we could use to help sketch the graph of a function, and the final one we would examine would be asymptotes. We introduced the horizontal asymptote.
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. 10 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise4.pdf

Other Exercise Sheets – Questions on the Second Derivative Test and Asymptotes

Section 4 Q. 4-5 from Problems

Past Exam Papers

Stationary Points are points a\in\mathbb{R} where the derivative of a differentiable function f:\mathbb{R}\rightarrow\mathbb{R}, f'(a)=0.

When asked to find the critical points of a function defined on the entire real line (rather than just on a closed interval [a,b]), the ‘endpoints’, \pm\infty are not considered critical points.

Convex is concave up and concave is concave down.

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 5 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 3(b), 5  from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 5(b),  from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 4(b),5(b), 6(a)  from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 5(b), 6(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2002/Maths_Stds/ms2001.pdf

Q. 4(a) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 4(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf


From the Class

1. Prove Theorem 5.2.2 (b)

MS 2001: Additions to Wills’ Notes

November 23, 2010 in MS 2001 | Leave a comment

The following topics are not covered in Wills’ notes:

  • Closed Interval Method (note that Wills does define Critical points – however we define critical points on closed intervals and include the endpoints)
  • First Derivative Test
  • Asymptotes

Leaving Cert: Integration

November 22, 2010 in General, Grinds, Leaving Cert, MA 1008 | Leave a comment

A short note covering integration for Leaving Cert maths.

 

(Please note that the proof of the Fundamental Theorem of Calculus inside isn’t quite correct. We need the Mean Value Theorem to prove it but the one in here is just for illustrative purposes.)

West Kerry Live Problem 1

November 22, 2010 in General, Leaving Cert | 2 comments

The Binomial Theorem is easier and more naturally proven in a combinatorics context but can be proven by induction.

Problem: Prove the Binomial Theorem by Induction.

Solution: Let P(n) be the proposition that for x,y\in\mathbb{R}, n\in\mathbb{N}

(x+y)^n=\sum_{i=0}^n{n\choose i}x^{n-i}y^i

(Binomial Theorem)

P(1):

\sum_{i=0}^1{1\choose i}x^{1-i}y^i={1\choose 0}x^{1-0}y^0=x+y=(x+y)^1

(P(1) is true)

Now assume P(k) is true; that is:

(x+y)^k=\sum_{i=0}^k{k\choose i}x^{k-i}y^i

Now

(x+y)^{k+1}=(x+y)^k(x+y)

=\left(\sum_{i=0}^k{k\choose i}x^{k-i}y^i\right)(x+y)

=\underbrace{\left(\sum_{i=0}^k{k\choose i}x^{k+1-i}y^i\right)}_{=S_1}+\underbrace{\left(\sum_{i=0}^k{k\choose i}x^{k-i}y^{i+1}\right)}_{=S_2}

Now all terms are of the form c(i)x^{k+1-i}y^i as i runs from 0\rightarrow k+1. Let j\in\{0,1,\dots,k+1\}. Now the x^{k+1-j}y^j term has constant from S_1 and S_2:

x^{k+1-j}y^j\left[{k\choose j}+{k\choose j-1}\right]

\Rightarrow (x+y)^{k+1}=\sum_{i=0}^{k+1}\left[{k\choose j}+{k\choose j-1}\right]x^{k+1-j}y^j

It is a straightforward exercise to show:

{n+1\choose k}={n\choose k}+{n\choose k-1}

Hence

(x+y)^{k+1}=S_1+S_2=\sum_{i=0}^{k+1}{k+1\choose i}x^{k+1-i}y^ii

(P(k+1) is true)

P(1) is true. P(k)\Rightarrow P(k+1). Hence P(n) is true for all n\in\mathbb{N}; i.e. the Binomial Theorem is true \bullet

MS 2001: Test 2 Sample

November 22, 2010 in MS 2001 | 2 comments

Ye have a test Wednesday 08/12/10. Please find attached a Sample

I will not be providing solutions to this sample. If you want solutions please attend the tutorial and ask me to do a question from the sample test.

Everything covered between Test 1 and the end of the year (01/12/10), except applied maximum/ minimum problems and some of curve-sketching (i.e. everything up to definition 4.5 in Wills’ notes MS2001. ) will be examinable. Essentially everything we did from Week 5 to this Monday 22/11/10 inclusive.

Question 1 (a) will have the same format as the sample, Question 1(b) will be taken from the exercise sheets, Question 2 from a past exam paper, and Question 3 will be on definitions & theorems (presented in class and also in Wills’ notes.

Research Update 6

November 22, 2010 in Research, Research Updates | Leave a comment

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I have finished off my revision of sections 1.2 (The Spectrum and the Spectral Radius) & 1.3 (The Gelfand Representation). Section 1.4 is a new topic for me – Compact and Fredholm Operators. A linear map T:X\rightarrow Y between Banach spaces is compact if T(B_1^X[0]) is totally bounded. As a corollary, all linear maps on finite dimensional spaces are compact. The transpose T^*:Y^*\rightarrow X^* has been introduced by Murphy is this chapter, and I have seen that if T is compact, then so is T^*. A linear map T is Fredholm if the T(X) and \text{ker }T are finite dimensional. In terms of progress, I am on p.25 of 265.

MS 2001: Week 9

November 17, 2010 in MS 2001 | Leave a comment

Firstly; there will be no MS 2001 lecture on Monday 6 December at 3 p.m. Instead you will have an MS 2003 lecture at this time in WG G 08. The 12 p.m. MS 2003 lecture on Wednesday December 1 in WG G 08 will now be an MS 2001 lecture.   Indeed it will be the final MS 2001 lecture as Wednesday 8 December is a test day and the week after is review week. The morning lecture at 9 a.m. on Wednesday 1 December will still go ahead.
On Monday we finished off the section on Implicit Differentiation.  We did the example of a circle and emphasised the use of the chain and product rules in this area. We used implicit differentiation techniques to establish the power rule:
\frac{d}{dx}x^n=nx^{n-1}
for n\in\mathbb{Q}, x\in(0,\infty), thus extending the rule we proved for integers.  We started a new chapter – Curve Sketching and Max/ Min Problems. We defined local maximum/ minimum and proved that if a function, continuous on a closed interval, takes an absolute max/ min at a point inside the interval, and is differentiable there, then the derivative must be zero.
In the tutorial we outlined Exercise Sheet 3, Q. 3(i)-(v) by stating where the functions were differentiable and what rule could be used to find the derivative where differentiable. We did Q. 4(i), Q. 10(i),(ii) and finally Q. 12. With a test in three weeks ye need to keep up the work on exercises.
On Wednesday we introduced critical points, the Closed Interval Method and the Second Derivative Test.
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

Use the Closed Interval Method to do Q. 8 (i), (iii) from

http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

Q. 14-17 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise3.pdf

Q. 2 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise4.pdf

Other Exercise Sheets

Section 4 Q. 1-3 from Problems

Past Exam Papers

Those questions in bold are to be done using the Closed Interval Method. Those questions in italic request the critical points of a function f:\mathbb{R}\rightarrow \mathbb{R} rather than f:[a,b]\rightarrow \mathbb{R}. In these questions the ‘endpoints’ \pm\infty are not considered critical points.

Q. 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 2, 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

Q. 2(ii), 4 from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 4(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 4(b), 5 from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 5(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 4(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 3, 4(a), 5(b), 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 5(b), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 4(c) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 4(c), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

From the Class

1. Prove Proposition 5.1.1 in the case that of x_1 is an absolute minimum.

Initial Topologies

November 16, 2010 in Research | Leave a comment

As part of my research of http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I came across a myriad of different topologies that could be put on various spaces associated with a space X. This piece is my attempt to collate and organise them. All are examples of initial topologies. An initial topology is determined by a family of functions.

Initial Topology (Belton 1.24)

Let X be a set and F be collection of functions on X, such that f : X \rightarrow Y_f , where (Y_f,\tau_f) is a topological space, for all f \in F. The initial topology generated by F, denoted by \tau_F , is the coarsest topology such that each function f\in F is continuous.  It is clear that \tau_F is the intersection of all topologies on X that contain

\bigcup_{f\in F} f^{-1}(\tau_f)=\{f^{-1}(U):f\in F,\,U\in \tau_f\}

Product Topology (Belton 3.20)

Let \{(X_a, \tau_a) : a \in A\}be a collection of topological spaces. Their topological product is (X, T), where (\sum is Cartesian product)

X =\sum_{a\in A} X_a:=\{(x_a)_{a\in A}: x_a\in X_a\,,\,\forall\,a\in A\}

is the Cartesian product of the sets X_a and \tau is the initial topology generated by the projection maps, for b\in A :

\pi_b:X\rightarrow X_b;(x_a)_{a\in A}\mapsto x_b

Strong Operator Topology (Belton 2.22)

Let X and Y be normed spaces; the initial topology on the bounded linear operators X\rightarrow Y, B(X, Y) generated by the family of maps \{T \mapsto Tx : x \in X\} (where Y is equipped with its norm topology) is called the strong operator topology.

Weak Topology (Belton 3.4)

Any normed space X gains a natural topology from its dual space, its weak topology. This is the initial topology generated by X^*, i.e., the coarsest topology to make each map \varphi\in X^* continuous. The weak topology on X is denoted by \sigma(X,X^*).

Weak* Topology (Belton 3.19)

Let X be a normed vector space. The weak* topology on X^* is the initial topology generated by the maps \hat{x}: X^* \rightarrow \mathbb{F}; \varphi \mapsto \varphi(x),\,\, (x \in X),i.e., the coarsest topology to make these maps continuous. The weak* topology on X^* is denoted by \sigma(X^*,X).

Research Update 5

November 15, 2010 in Research, Research Updates | Leave a comment

With a bout of illness last week I only got to finish off Beltonhttp://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf and start Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

At present I am speedily going through Chapter 1: Elementary Spectral Theory. This has all been done in Belton but I like Murphy’s lucid and no-frills approach. In places Murphy takes a different approach to Belton (e.g. the proof that the spectrum is non-empty establishes the differentiability of the map \mathbb{C}\backslash \sigma(a)\rightarrow A, \lambda\mapsto (a-\lambda)^{-1} without recourse to the resolvent). This quick revision will continue until 1.4 Compact and Fredholm Operators – which is a new topic for me. In terms of progress, this starts on p.18; I’m presently on p.9. The entire book weighs in at 245 pages and realistically I certainly wouldn’t expect to be finished before Christmas.