## Assessment 2

Will be held on the Friday of Week 11 (24 November), at 09:00 in B214 (the usual lecture venue). A sample has been emailed and I also have hard copies with me. I strongly advise you that attending the tutorial alone may not be sufficient preparation for this test so you may have to devote extra time outside classes to study.

## Week 10

In Week 10 explored Network Theory, or rather Graph Theory, in more depth. We looked at digraphs, connectedness, valency, walks, and trees.

## Week 11

We will finish our study of Graph Theory by looking at Eulerian graphs, Fleury’s Algorithm, Hamiltonian graphs, and Dirac’s Theorem. We will then begin the last chapter on recursion. We have our test on Friday

## Week 12

In Week 12 we will finish our study of recursion and perhaps do a little revision.

## Week 13

In Week 13, perhaps we will have five tutorials (normal rooms and times) of which you are invited to up to four (your own tutorial slot plus the up to three of the lecture slots).

## Study

Some students need to do extra work outside tutorials. Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

## Geometric Series

Let $a,r\in\mathbb{R}$ be constants. Let $\{a_n\}$ be a sequence of real numbers with the following recursive definition:

$a_n=\begin{cases}a & \text{ if }n=1\\ r\cdot a_{n-1}&\text{ if }n>1\end{cases}$.

Therefore the sequence is given by:

$a,ar,ar^2,ar^3,ar^4,\dots$

Such a sequence is called a geometric sequence with common ratio $r$.

When we add up the terms a sequence we have a geometric sum:

$S_n=a+ar+ar^2+ar^3+\cdots ar^{n-1}$.

Here $S_n$ is the sum of the first $n$ terms.

We can find a formula for $S_n$ using the following ‘trick’:

$r\cdot S_n=ar+ar^2+ar^3+\cdots ar^n$

$\Rightarrow a+r\cdot S_n-ar^n=S_n$

$\Rightarrow S_n(r-1)=a(r^n-1)$

$\displaystyle \Rightarrow S_n=\frac{a(r^n-1)}{r-1}$.

### Exercises

Assuming that $|r|<1$, find a formula for the geometric series

$\displaystyle S_{\infty}=\lim_{n\rightarrow \infty}S_n$.

## Binary Numbers

### Exercises

• Write the following as fractions:

$0.1_2,\,0.11_2,\,0.101_2$.

• Use infinite geometric series to show that:
• $0.111\dots_2=1$
• $0.0111\dots_2=\frac12$
• $0.101010\dots_2=\frac23$

## Doubling Mapping

The doubling mapping $D:[0,1)\rightarrow [0,1)$ is given by:

$\displaystyle D(x)=\begin{cases}2x & \text{ if }x<1/2 \\ 2x-1 & \text{ of }x\geq 1/2\end{cases}$.

### Exercises

• Find the first six iterates of the point $x_0=\frac17$ under $D$.
• Find the first four iterates of the point

$x_0=\frac{1}{2}+\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}=0.1101_2$.

• Where $x$ has the binary representation

$x = 0.a_1a_2a_3a_4a_5a_6a_7a_8\dots$ ,

write down expressions for $D(x)$ and $D^5(x)$.

• Hence find points $y, z \in [0, 1]$ such that $y$ and $z$ agree to 5 binary digits but $D^N(y)$ and $D^N(z)$ differ in the first binary digit for some $N \in \mathbb{N}$.
• Describe the period-5 points of $D$.
• Let $w \in [0, 1]$ have a binary representation beginning $w = 0.01001\dots$  . Find a period-5 point $\gamma$ of $D$ such that $w$ and $\gamma$ agree to five binary digits.
• Find a $\delta \in [0, 1]$ such that there are iterates of $\delta$, $D^{n_1}(\delta),D^{n_2}(\delta),D^{n_3}(\delta)$, with $n_1, n_2, n_3 \in \mathbb{N}$, that agree with 0.111 , 0.101, and 0.010, to three binary
digits.

## Sensitivity to Initial Conditions

### Exercise

Let $f(x)=4x\cdot (1-x)$. Where $[0,1]$ is the set of states, and $f:[0,1]\rightarrow [0,1]$ the iterator function, by looking at the first seven iterates of $x_0=0.8$ and $y_0=0.81$, show that this dynamical system displays sensitivity to initial conditions [HINT:4*ANS*(1-ANS)]

## Test 2

Thursday 23 November at 09:00 in the usual lecture venue. You will be given a copy of these tables. Based on Chapter 3, samples at the back of Chapter 3 and also here (Q. 4 has a typo — it should be $e^{-x}\sin(y)$). I strongly advise you that attending tutorials alone will not be sufficient preparation for this test and you will have to devote extra time outside classes to study aka do exercises.

## Week 10

We started Chapter 4 by looking at integration by parts. We started looking at completing the square.

## Week 11

We will look at completing the square and work.

## Dynamical Systems

A dynamical system is a set of states $S$ together with an iterator function $f:S\rightarrow S$ which is used to determine the next state of a system in terms of the previous state. For example, if $x_0\in S$ is the initial state, the subsequent states are given by:

$x_1=f(x_0)$,

$x_2=f(x_1)=f(f(x_0))=(f\circ f)(x_0)=:f^2(x_0)$

$x_3=f(x_2)=f(f^2(x_0))=f^3(x_0)$,

and in general, the next state is got by applying the iterator function:

$x_{i}=f(x_{i-1})=f^i(x_0)$.

The sequence of states

$\{x_0,x_1,x_2,\dots\}$

is known as the orbit of $x_0$ and the $x_i$ are known as the iterates.

Such dynamical systems are completely deterministic: if you know the state at any time you know it at all subsequent times. Also, if a state is repeated, for example:

$\text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_4=x_2,x_5\dots,\}$

then the orbit is destined to repeated forever because

$x_5=f(x_4)=f(x_2)=x_3$,

$x_6=f(x_5)=f(x_3)=x_4=x_2$, etc:

$\Rightarrow \text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_2,x_3,x_2,\dots\}$

### Example: Savings

Suppose you save in a bank, where monthly you receive $0.1\%=0.001$ interest and you throw in $50$ per month, starting on the day you open the account.

This can be modeled as a dynamical system.

Let $S=\mathbb{R}$ be the set of euro amounts. The initial amount of savings is $x_0=50$. After one month you get interest on this: $0.001\times50$, you still have your original $50$ and you are depositing a further €50, so the state of your savings, after one month, is given by:

$x_1=50+0.001\times 50+50=(1+0.001)50+50$.

Now, in the second month, there is interest on all this:

interest in second month $0.001\times((1+0.001)50+50)=0.001x_1$,

we also have the $x_1=(1+0.001)50+50$ from the previous month and we are throwing in an extra €50 so now the state of your savings, after two months, is:

$x_2=x_1+0.001x_1+50=(1+0.001)x_1+50$,

and it shouldn’t be too difficult to see that how you get from $x_i\longrightarrow x_{i+1}$ is by applying the function:

$f(x)=(1+0.001)x+50$.

#### Exercise

Use geometric series to find a formula for $x_n$.

## Weather

If quantum effects are neglected, then weather is a deterministic system. This means that if we know the exact state of the weather at a certain instant (we can even think of the state of the universe – variations in the sun affecting the weather, etc), then we can calculate the state of the weather at all subsequent times.

This means that if we know everything about the state of the weather today at 12 noon, then we know what the weather will be at 12 noon tomorrow…

The following question gave a little grief:

Two smooth spheres of masses $2m$ and $3m$ respectively lie on a smooth horizontal table.

The spheres are projected towards each other with speeds $4u$ and $u$ respectively.

i. Find the speed of each sphere after the collision in terms of $e$, the coefficient of restitution

ii. Show that the spheres will move in opposite directions after the collision if $e>\frac13$.

My contention is that the question erred in not specifying that the answers to part i. were to be in terms of $e$ and $u$.

Solution:

i. The following is the situation:

Let $v_1$ and $v_2$ be the velocities of the smaller respectively larger sphere after collision. Note that the initial velocity of the larger sphere is minus $u$.

Using conservation of momentum,

$m_1u_1+m_2u_2=m_1v_1+m_2v_2$

$\Rightarrow 2m(4u)+3m(-u)=2mv_1+3mv_2$

$\Rightarrow 5u=2v_1+3v_2$.

Using:

$\displaystyle e(u_1-u_2)=-(v_1-v_2)\Rightarrow \frac{v_1-v_2}{u_1-u_2}=-e$,

Therefore,

$\displaystyle \frac{v_1-v_2}{4u-(-u)}=-e$

$\Rightarrow v_1-v_2=-5ue\Rightarrow v_1=v_2-5ue$,

and so

$5u=2(v_2-5ue)+3v_2\Rightarrow 5u=2v_2-10ue+3v_2$,

$\Rightarrow 5v_2=5u+10ue\Rightarrow v_2=u(1+2e)$.

$\Rightarrow v_1=u(1+2e)-5ue=u+2ue-5ue=u(1-3e)$.

ii. $v_2>0$. If $e>\frac13\Rightarrow 3e>1\Rightarrow 1-3e<0$ and so

$v_1=u(1-3e)<0$;

that is the particles move in opposite directions.

## Week 9

In Week 9, will looked at examples of functions, including lines, quadratic functions, polynomial functions, exponential functions, the natural logarithm function, the floor function, and the ceiling function. We began the chapter on Network Theory by looking at the Bridges of Konigsberg Problem.

## Week 10

In Week 10 we will explore Network Theory, or rather Graph Theory, in more depth. We will look at digraphs, connectedness, valency, walks, and trees.

## Assessment 2

Will be held on the Friday of Week 11 (24 November), at 09:00 in B214 (the usual lecture venue). Expect a sample test shortly.

## Study

Some students need to do extra work outside tutorials. Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

## Distances between Probability Measures

Let $G$ be a finite quantum group and $M_p(G)$ be the set of states on the $\mathrm{C}^\ast$-algebra $F(G)$.

The algebra $F(G)$ has an invariant state $\int_G\in\mathbb{C}G=F(G)^\ast$, the dual space of $F(G)$.

Define a (bijective) map $\mathcal{F}:F(G)\rightarrow \mathbb{C}G$, by

$\displaystyle \mathcal{F}(a)b=\int_G ba$,

for $a,b\in F(G)$.

Then, where $\|\cdot\|_1^{F(G)}=\int_G|\cdot|$ and $\|\cdot\|_\infty^{F(G)}=\|\cdot\|_{\text{op}}$, define the total variation distance between states $\nu,\mu\in M_p(G)$ by

$\displaystyle \|\nu-\mu\|=\frac12 \|\mathcal{F}^{-1}(\nu-\mu)\|_1^{F(G)}$.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative $\mathcal{L}^p$ machinary shows that:

$\displaystyle \|\nu-\mu\|=\sup_{\phi\in F(G):\|\phi\|_\infty^{F(G)}\leq 1}\frac12|\nu(\phi)-\mu(\phi)|$.

(supremum presentation)

In the classical case, using the test function $\phi=2\mathbf{1}_S-\mathbf{1}_G$, where $S=\{\nu\geq \mu\}$, we have the probabilists’ preferred definition of total variation distance:

$\displaystyle \|\nu-\mu\|_{\text{TV}}=\sup_{S\subset G}|\nu(\mathbf{1}_S)-\mu(\mathbf{1}_S)|=\sup_{S\subset G}|\nu(S)-\mu(S)|$.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in $F(G)$, and therefore the classical total variation distance is equal to:

$\displaystyle \|\nu-\mu\|_P=\sup_{p\text{ a projection}}|\nu(p)-\mu(p)|$.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance $\neq$ Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a $M_n(\mathbb{C})$ factor of $F(G)$ might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

## Test 2

Tuesday 21 November Thursday 23 November at 09:00 in the usual lecture venue. You will be given a copy of these tables. Based on Chapter 3, sample at back of Chapter 3. I strongly advise you that attending tutorials alone will not be sufficient preparation for this test and you will have to devote extra time outside classes to study aka do exercises.

## Week 9

We finished looking at Partial Differentiation and then saw how it can be used in error analysis.

## Week 10

We will start Chapter 4 by looking at integration by parts. We might look at completing the square and work.

## Assessment 2

Deadline 16:00, Monday 20 November, Week 11. Note that when you open MATH7019A2 – Student Data, you should see a list of numbers that you are supposed to use in the questions. All of the $w_i,\,L_i,\,a,\,b$ are to be taken as these constants. It is only $E$ and $I$ that are to be kept as ‘free variables’.

## Week 9

We finished looking at the normal distribution and then looked at Sampling Theory.

## Week 10

We will look at Hypothesis Testing and begin Chapter 4 with a Revision of Differentiation.

## Week 8

In Week 8, we lost another lecture with the bank holiday, but we had an opportunity to better understand the function concepts after looking at arrow diagrams and the graph of a function.

## Week 9

In Week 9, we will look at examples of functions, including lines, quadratic functions, polynomial functions, exponential functions, the natural logarithm function, the floor function, and the ceiling function.

## Test 1

Results have been emailed to you. Solutions and comments here.

## Assessment 2

Will be held in Week 11. Proper notice in Week 9, and a sample in Week 10.

## Study

Some students need to do extra work outside tutorials. Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

Anyone who is missing notes is to email me.

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

 J.P. McCarthy on MATH6040: Winter 2017, Week… Student on MATH6040: Winter 2017, Week… J.P. McCarthy on MATH6040: Winter 2017, Week… Student on MATH6040: Winter 2017, Week… MATH6028: Chaos Theo… on MATH6028: Chaos Theory (Part…