Branch Cuts Explained in Ballerina Turns

Suppose you are standing up and you look at a green wall. Also, you are a ballerina or just feel like doing a full 360 degrees (or  ) turn. Naturally after your turn, you should see the same green wall, yes?

The answer (generally) is no. It all depends on what you can see! It also depends on how you are looking.

Suppose that the wall takes two colours: green and blue. They are definitely not the same!

This should appear the same after a turn…

Mathematics can be bizarre and sometimes things that do not appear to be the same actually… are. Suppose we are not in such a universe. Green and blue are definitely different for us.

We get a blue wall by using blue paint to paint the wall blue. Similarly we get a green wall by using green paint.

Then we can say dependent on what paint we used, we get a different output.

What if in our seemingly normal mathematical universe we use green paint to paint the wall green, look at the wall standing up, then do a full turn and the wall is now blue?

And further suppose that this can happen an infinite amount of times?!

But we did not use blue paint!!

Painting something is a single act. You do something and you get one (and only one) thing out in return, which makes sense.

But looking back at what you paint is not the same! Because you can paint with two different colours, when you look back there are two possible colours! You could call this a multi-act.

Also suppose there is not a single way to just look. Maybe we can look with squinted eyes or with just one eye, the other closed, or enclosed: pirate style!

Now this is crucial: how we look could affect what colour we see.

Suppose there is some way of looking that, if we make the look, then go to the wall and “look” back at the look, we can only see one thing: the look.

This appears silly, of course that’s true. But also suppose we can look, go to the wall and “look” back and see the same and perhaps another look. You could call this a 2-look.

Generalise it further and you have a multi-look. A capability to look back at your projection and see something else.

Then if you can look forward, do a turn, say a 1-turn, generalised to multi-turns, you should always see a green wall provided that you painted it green initially.

But if the “look” that you are using is a multi-look, clearly when you look back, which if you think about it… is just the same as doing a turn, then you don’t have the green wall. You have a blue wall.

This is precisely what a branch point is: the “look” that you choose that after a “turn” gives you a different colour to the wall.

To fix this, we could ban certain looks (so all multi looks) and focus on the single looks: these are all continuous since no matter how many turns we do, we have the same green paint always appearing.

Then if consider the multi looks, we make these continuous by fixing your turns: where you start and where you finish. You should still do a full turn either way. You just change the position where you do it.

This means if we are in the room and the paints are at some position in the room, so the green paint at position A and blue paint at position B, consider the distance between them, call it C.

This C will be removed: from blue a turn gives us green and likewise from green to blue. Formally, we take a cut in the axis between the points A and B.

This is a branch cut.

This part is removed as travelling in this small part of the considered axis gives us a discontinuity! Everything else stays!

Now suppose we could transfer this to a different room with different paint colours and different representations of what gives us the same or different (so in our considered case, the representation is looking at the wall).

This is analytic continuation.

This may not generally work: other rooms and other representations may have a specific structure which mean what is discontinuous for one is not discontinuous for the other.

Then suppose when in some specific section (a contour) of the room, everything is well defined and after every single turn we have the same green wall.

Suppose we are a different section and given the “nice” behaviour of the first section, if we can somehow exhibit the same behaviour, we have connected two sections.

Then we may do this for the whole room and hence when we look and turn, we always see a green wall.

This is the Monodromy Theorem.

Then we understand the whole room, after our ballerina turns, anywhere in the room, we always see green,

Essentially, we have just explained a bunch of useful (and closely related) concepts and theorems in complex analysis.

We weren’t being very formal – far from it, but we have the idea in our mind.

Formally, we looked at multivalued functions and their relationship as the inverse of single-valued functions.

We see what happens as we travel through a full period: the multivalued function returns a different value at some specific points (the branch points).

Then we fix angles and remove parts of the imaginary or real axis: a branch cut.

Whether this cut holds in a different region and whether there is any difference to how we look at the wall or the colour of the wall is analytic continuation.

An example of analytic continuation..

Finally, the Monodromy Theorem characterises this with singular points and analyticity.

Related links:

Branch cuts, points
Multivalued functions
Analyticity
Analytic continuation
Monodromy theorem

Partitions: The Most Powerful Tool In Mathematics

If you have a big object and cannot understand it, what do you do? You break the big object into little objects, try to make sense of them and then build up back to the big object again.

These little objects are partitions of the big object.

Let us see how this tool has been used in some areas of mathematics:

Integration (Analysis)

The first and original integral, the Riemann integral uses partitions of a set and then creates specific sums (Riemann Sum) of these partitions which satisfy certain criterion for integrability.

The result? The integral and in general integration and of course The Fundamental Theorem Of Calculus.

Group Theory (Algebra)

Suppose you have a group. Represent this group as a big box. The things inside this big box clearly make it what it is: the big box.

Partition these things inside into sets and if they have some structure (subgroups), collect all of them and consider the divisibility of the cardinality of the partitions.

What is inside one box may not be what is inside another box (in group talk: a left coset is not necessarily equal to the right coset associated to the group)

The result? Lagrange’s theorem (there are other results but the sheer beauty of that one should be enough)

Graph Theory (Algebra)

Suppose you have a graph. Partition the graph into something smaller: a collection of subgraphs. If this collection is a disjoint union of n-partite graphs, we can understand the graph’s structure.

A specific structure will allow us to understand if the graph is planar and to resolve key, real-life problems such as the utilities problem.

The result: Euler’s characteristic, Four-Colour Theorem, Perron-Frobenius Theorem.

Stochastic Processes (Probability)

Suppose you are doing some event B contained in a sample space W. But to do B you must “be” in some place, say, A. Maybe you are in A_0, A_1, A_2, … and so on. The point is, you must be there. You partition the chance of event B occurring as sum of the chance of B occurring given that you are in A multiplied independently by the chance of you being in A.

This is the Law Of Total Probability.

The result: Chapman-Kolmogorov equation.

Actually this theorem can be represented into a matrix form which then allows us to produce big theorems for Markov chains and (in general) Markov processes.

The result: Markov chains, Ergodic theorem

Even more, the whole of probability theory arises from this.

Introductory stochastic processes classes look at how some initial understanding of probability (and other basic analysis modules) allows us to understand the world: reliability theory (how likely is it that something will break down?), queuing theory (what is that chance of you waiting for a specific time period in a given queue at say, the supermarket, before you go to the till?) and so on.

Graduate classes then use the idea of partitions and breaking things down so much that it links nicely with the “rigorous” definition of probability: measure theory.

These are just some applications, the explanations are not detailed or interesting enough to explain why using partitions is so crucial in all various fields of mathematics.

The point: if you do not understand something, break it down into what you can (or will) understand. Then collect these little pieces together and see what you get.

Supermarkets puzzle

Supermarkets only have decorations during December. Ten times as many people visit in December than any other month. You enter the supermarket, what is the probability that you see Christmas decorations?

Transformers 3 Puzzle

The movie Transformers: Dark of the Moon contains an interesting puzzle.

Sam, the main character, has an evil robot (a small Decepticon) which transforms to a watch placed on his hand by Dylan, an evil guy.

The robot/watch can understand everything Sam says and hears. It reports information back to Dylan. It can tap into Sam’s nervous system and affect him, meaning he cannot just remove it. Sam must communicate to fellow humans and good robots (Autobot’s) but he knows the whole time whatever he hears or says is accessed to the enemy.

We can build an interesting question from this.

Suppose you know person X and Y, person X works for a competing, evil company X’ and person Y works for your employing, good company Y’. Suppose everything that you communicate with person Y goes to person X and you cannot inform person Y of this.

Devise a winning strategy to defeat company X’ and for company Y’ to win.

Related mathematical fields; algebra, probability theory, set theory, number theory.

What is a good mathematical question?

What does a mathematician do? A good question.

One could say a mathematician observes and provides a logical input.

Another could say that mathematicians prove theorems.

And yet, another, could say that mathematicians solve problems.

The first is a defining feature of an even bigger question (what is mathematics?), occurring prior (and hence leading) to axiomatizing.

The second uses axioms to produce facts.

The third is to apply the first two, providing a desired answer to some question.

This post looks at what that question is. What it should be.

Formally, we are asking, what is a good mathematical question?

(We could quip in by saying, is that (and hence this) question a good mathematical question itself?)

Mathematicians spend so much of their time solving questions. Knowledge of what a good question is should help us to solve the question, independent of the question itself.

In fact, this is the point of mathematics; to provide factual statements that are independent of who is providing them.

What is a good mathematical question?

We could say one that is clear and has a purpose i.e. there is no ambiguity to solving

However, a complicated question can be asked. Life is complicated, often asking questions that seem nonsensical. Therefore, our first question is is not sufficient, certainly it may be a good question, but better can be asked.

A good question can be difficult. Sir Andrew Wiles spent some years proving the Taniyama-Shimura conjecture, which asked several difficult questions.

Some thought it would take a long time to prove as we did not know where to begin to understand the questions let alone to solve it. One man took his time to understand and as a result, we all do.

Sir Andrew Wiles, solving a problem.

Understanding the question is the initial process. You learn the motivation of the question, the required theory. Then, solving the question requires applying theory and mathematical implementation of ideas gained from understanding.

We may conclude with a good question is one that takes some time and careful thinking to solve.

But still, this is not enough. Some complicated questions require fuzzy statements (statements which have not been verified, do not exist and are not true in general and so on). Some complicated questions require numbers which we cannot fathom of.

Some need to be verified by a computer, as we cannot solve that specific part. Some complicated questions themselves are clear yet give way to fuzzy premises in an attempt to solve.

Some complicated questions are just too complicated. You can spend your whole life thinking about them, hopefully solving them. Ideally, a question is a gateway to understanding and gaining knowledge.

Few of us have the time Sir Andrew Wiles or Grigori Perelman did to solve complicated questions.

But these complicated questions are infinitely better than our easy questions. We give an example of easy questions and complicated questions.

Easy questions requires a little bit of time, some specific problem solving skills and a bit of knowledge of undergraduate mathematics.

Complicated questions require a lot of time, new problem solving skills and new mathematical knowledge.

Suppose we want to “solve” the continuum hypothesis. We would need to be comfortable with theories involved in set theory, logic and then learn the method of forcing as outlined by Paul Cohen.

With progress, we would solve several other (easier) questions which would give us more knowledge, independent of solving the original question.

There is a balance, it seems. Is a question asked with some clarity, yet with an interesting element enough?

The Goldbach Conjecture satisfies both of these criterions (clarity; it makes sense, interesting element; knowledge of primes), yet is a surprisingly (actually, unsurprisingly) complicated question, currently unsolved. Although it is of interest, there is no solution, all methods, techniques, ideas and such eventually fail.

Perhaps a good question is one that does not need to be answered. But then, why was it asked? To understand. If we understand, we will (hopefully) eventually solve.

In our modest world, it is fair, then, to say that a good question is one not simple enough to be solved in a couple of minutes; one which requires to carefully think. One requiring some non-trivial theory and to be reasonably solvable.

A question needing different theory to solve is even better.

Say we get asked to find the probability of something event happening, to do so we may need combinatorics to understand our specific event, some theorems regarding the factorial function.

Then we may need linear algebra to solve some linear equations for our distribution, then some knowledge of probability theory, perhaps knowledge of expectation, conditioning probability and such. Through this path, you see why the question was interesting; no clear path existed in solving it.

Eventually you build the intuition to create the clear paths to solve questions.

Such a question is good as you cannot cheat it. You need to think. The solution is not so obvious. A small change to the question can have no answer. But we understand. Then we move to the next question!

An Introduction To Stochastic Calculus

When performing integration, differentiation, evaluating sums and products; our evaluations use numbers (usually real, sometimes complex). They define the “answer” we get; a physical interpretation into understanding what we have done.

Suppose instead of numbers, we use variables. Specifically, we use random variables. Then we are using a new calculus; appropriately named stochastic calculus, stochastic meaning random.

This post is not going into probability theory nor will it define what a random variable is, specifically we introduce stochastic calculus without any extreme formality.

Suppose you want to look at the stock market, or anything that enough humans touch in response of being able to gain something in return, say an environment. Say we define to be the price of asset at time , then consider some time period , we ask several questions:

How small does dt need to be?
What happens when it’s infinitesimal?
Does it relate to some specific probability distribution?
Can we make a formula for X without involving any explicit, complicated functions?

First we continue with our asset, given a change in time, we have a change in price .

Suppose the asset’s environment is responsive to change and there are many users like us, the controller of the asset; all looking after their asset. Is the change in the asset price simply just how the market fluctuates at the given time, multiplied by some constant for normalisation?

Suppose this is true and thus we have the differential equation .

(where is the change in price of the asset at time , is a constant and is some bizarre function that can understand and display the change in market fluctuations and how the market, and generally environment, responds to our asset.

Integrating and using some initial condition (), we have a seemingly nice and a intuitive solution.

We have

For short time periods (specifically, for really short) it works fine. But it’s hugely problematic.

As time increases, the probability of the asset having a negative price is bigger than zero, ie non zero, therefore it can happen on some day, or specifically some time period. Formally we have the statement “ to be strictly bigger than zero" to be true.

Initially we think it’s not too probable but just that it exists gives us problems. Who’s going to trust or use some differential equation that says the probability of the asset having a negative price at some future time using that differential equation is non zero?

So we go back to our differential equation. Do we start over again? Not really, we just think sensibly. Suppose I have some hot tea that I want to drink it when it is just right. So I let it “play” in the environment and then make a judgement on whether I should drink it or not.

Relating to our problem, an investor or user of an asset will look at the potentials gain or loss (change) as opposed to the initial asset price, this shows that we think about not only the price, but how the price changes in the future, despite trying to answer the future.

So we have the relative price change to be the proportional to market fluctuations, we have

Now by integrating from the initial price to the variable , this gives us .

Surely we can solve this, right? Trying our usual elementary methods makes no sense since our function is the function that looks at market fluctuations – in mathematical terms; nowhere differentiable and non continuous, so we arre stuck.

Unless we define a new integral equation and a new kind of calculus; the counterpart to real numbers, one for random variables. We have stochastic calculus.

Questions:

What happens when the time increments approach zero? (Brownian Motion)

With limiting zero time increments do we have a differentiable market fluctuation function? What probability distributions do we use to understand how this process works? (Poisson process)

What new type of integral do we define to solve the weird integral we have on the right hand side? (Ito integral)

This new integral must apply to some kind of calculus given that we will be differentiating it at some point and other proper tires, specifically what is this calculus and what special defining properties does it have? (Stochastic calculus)

Here we stop given that the questions asked cover enough of the motivation to study stochastic calculus, however for further enthusiasts we ask more questions and give some new ideas.

Given a discrete random variable mapped by some specific distribution, in between time intervals we can actually use a continuous random variable to “travel” through the time increments. The related distributions are the Poisson and Exponential (and discrete analogy Geometric). How can we use these distributions to understand what is happening in our asset price change, or in our differential equation?

This function, supposedly non differentiable everywhere, having a limiting time increment to zero may change how it behaves. It may even make solving the DE easier, but there are problems with it. What are these problems? Why do we care for small time increments?

Stochastic Calculus requires good knowledge of probability theory, real analysis, linear algebra and a few other bachelor level modules (complex analysis, differential equations, Hilbert spaces) to fully let it sink in.

Lecture 1: Sets, Real Numbers, Fields, Ordered Fields

The real number system is a set on which the operations of addition and multiplication are defined so that every pair of real numbers (say and ) has a unique sum and product, both real numbers, with the following properties:

Given any (and all) , we have

(A) and (Commutative laws)

(B) and (Associative laws)

(C) (Distribute law)

(D) There exist distinct real numbers and such that and , for all .

(E) For each there is a real number such that and if , there is a real number such that

A set on which two operations are defined so as to have properties (A) – (E) is a field.

Motivation: Think about these properties for a second before moving on. (A) establishes what the operations do, (B) then asks what happens when we include more elements, a “double operation” if you like. (C) then mixes the two operands into one. (D) and (E) are closely related. Suppose we want to add an element to our given real number that… gives us our real number? So we are adding nothing, or . Suppose we have the same idea for our other operand, multiplication, we have the real number to do this for us.

Then we ask, can we find real numbers that give us what we just discovered, namely and from addition and multiplication? For that we have and with for the second condition.

Exercise: Can you think of another example of a field?
Exercise: Do the set of integers form a field? Verify through (A) – (E).

The real number system is ordered by the relation , which has the following properties:

(F) For each pair of real numbers and , exactly one of the following is true: , , .

(G) If and then . (Transitive)

(H) If , then for any and if then .

A field with an order relation satisfying (F) – (H) is an ordered field. The real numbers (and rationals) form an ordered field.

Suppose we get bored of the real numbers and define a new field satisfying everything does but is defined as such. For any we define a new number as , where . We call this set , the set of complex numbers.

Exercise: Verify that is a field.
Exercise: Is an ordered field? (Use (F) – (H) to prove order or give a counterexample)
(Hard) Exercise: Why have we defined and not ?

Questions to think about:

Are all fields ordered fields?
Are finite fields ordered?
What do these two above questions (if and when answered) say about ordered fields? (Hint: Infinite number of…)