Projection Distance Equal to Total Variation Distance?

November 9, 2017 in C*-Algebras & Operator Theory, Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research | 2 comments

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).

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MATH6040: Winter 2017, Week 9

November 9, 2017 in MATH6040 | 2 comments

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.

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MATH7019: Winter 2017, Week 9

November 8, 2017 in MATH7019 | Leave a comment

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.

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MATH6055: Winter 2017, Week 8

November 3, 2017 in MATH6055 | Leave a comment

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.

 

 

MATH6040: Winter 2017, Week 8

November 2, 2017 in MATH6040 | Leave a comment

Test 2

Tuesday 21 November at 09:00 in the usual lecture venue. 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 8

We missed a lecture with the bank holiday: we looked at Implicit Differentiation and  started Partial Differentiation.

Week 9

We will finish looking at Partial Differentiation and see how it can be used in error analysis.

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MATH7019: Winter 2017, Week 8

November 2, 2017 in MATH7019 | Leave a comment

Assessment 1

Apologies that it took until this week to get the results back to you. I will bring the assignments with me tomorrow (Friday 3 November, Week 8) for you to have a look at.

By and large I was happy with the work. I might, however, send out a warning. I think it has been the case that students do far better in Assessment 1 than they do in the Final Exam.

Some comments on common mistakes.

Assessment 2

Deadline 16:00, Monday 20 November, Week 11

Week 8

We missed the Monday Lecture with the bank holiday. On Wednesday we looked at the Poisson distribution and started talking about the Normal distribution.

 

Week 9

We will finish looking at the Normal distribution and then look at Sampling Theory.

 

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How does a Curve have an Equation?

November 1, 2017 in A1 Leaving Cert Maths, General, MATH6015, MATH6040, MATH6055 | 8 comments

In school, we learn how a line has an equation… and a circle has an equation… what does this mean?

The short answer is

points (x_0,y_0) on curve \longleftrightarrow solutions (x_0,y_0) of equation

however this note explains all of this from first principles, with a particular emphasis on the set-theoretic fundamentals.

Set Theory

A set is a collection of objects. The objects of a set are referred to as the elements or members and if we can list the elements we include them in curly-brackets. For example, call by S the set of whole numbers (strictly) between two and nine. This set is denoted by

S=\{3,4,5,6,7,8\}.

We indicate that an object x is an element of a set X by writing x\in X, said, x in X or x is an element of X. We use the symbol \not\in to indicate non-membership. For example, 2\not\in S.

Elements are not duplicated and the order doesn’t matter. For example:

\{x,x,y\}=\{x,y\}=\{y,x\}.

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MATH6040: Winter 2017, Week 7

October 26, 2017 in MATH6040 | Leave a comment

Catch Up Material

If you have not already, you should watch:

Week 7

We looked at Parametric Differentiation and Related Rates

Week 8

We will look at Implicit Differentiation and Partial Differentiation.

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MATH7019: Winter 2017, Week 7

October 25, 2017 in MATH7019 | Leave a comment

Catch-Up Material

Please watch this cantilever example and this summary of beams if you have not done so already.

Week 7

We finished looking at Chapter 2 by looking at the Three Term Taylor Method for approximating solutions of ordinary differential equations.

On Wednesday we started Chapter 3 (Probability and Statistics) by looking at some general concepts in probability and then we looked at random variables with a binomial distribution.

We lose a tutorial on Friday with the day off.

Week 8

We miss another lecture on Monday with the bank holiday. Wednesday, we will look at the Poisson distribution and perhaps the Normal distribution.

Assessment 1

I would have hoped to have finished the corrections by now however this has been delayed by a number of factors. My hope would be to have the results for you next week. Apologies for this delay.

Assessment 2

Deadline 16:00, Monday 20 November, Week 11

Study

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.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH6055: Winter 2017, Week 7

October 25, 2017 in MATH6055 | 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 jpmccarthymaths@gmail.com and I will add you to the mailing list.

Catch-Up Material

If you haven’t watched these at this stage you really should:

Week 7

Disrupted by the Friday off, we did two lectures on Functions, looking at the definition of a function. We looked at what it means for a function to be 1-1, onto, and bijective. We looked at the composition of two functions as well as the concept of an inverse function.

Week 8

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

Assessment 1

Results have been emailed to you. I will send ye on a copy in a week or two.

Assessment 2

Will be held in Week 11.

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.