Ramsey, Parties, and The Probabilistic Method

Frank Ramsey is all kinds of awesome. A mathematician who died far too young … but honestly, I really can’t begin a post like this without mentioning the fact that Frank Ramsey’s brother became the Archbishop of Canterbury.

I’m glad I got that out of the way.

Now, Ramsey is responsible for this branch of discrete math that is best exampled I think by the following problem – it also serves as a very nice example of the probabilistic method.

To begin with – next time you’re at a party or a bit of a get together, it probably goes down something like this. You walk in, say hello to the host, grab some drinks, wander, pet the dog, smile at people you think you’re supposed to know, brush the dog hair off your pants, then you see a group of people you know, and you wander over and chat with them for a bit. Note though that this is a mathematically ideal party, in that we assume if you know someone, they also know you : )

Of course, if you look around, you’re likely to see many such groups, of varying sizes, groups of mutual friends all saying nasty things about the people in your group.

Cliques, in a word.

The issue at hand is the -size- of the groups.

A Brief Jaunt Towards Rings

The post probably ought to have a warning of some sort, but I can’t imagine what that would be. “WARNING: Algebra!” “Caution: Abstraction Ahead!” Judging by the content you are about to bear witness to, it should likely read something like “DANGER: Sleepily Cognated Flunctions Inverted Paramour In Delhi.”

Warning: Long, Rambling Exposition of Rings and Ring Theory Ahead

A New Year

First day of classes, that was. Always a good one. A new year, a new semester : ) I am excited. I always am. Hopeless optimist, that’s me – even on my most depressed of days. A good day over all. Got back into the swing of things with a dose of Quantum Mechanics this morning, which I think I am going to enjoy sharing here. Followed that up with a bracing hit of Real Analysis, more specifically functions of a bounded stretch. Not sure I’m going to talk much about analysis. It just doesn’t appeal much to me. Followed that up with a break (something I didn’t get last semester, what with 6 hour blocks of classes), then some Differential Equations, to knock that requirement out, and then finished off with another dash of Real Analysis. The other spot of fun this evening was a talk on Ruby on Rails, which should prove to be an interesting course.

Nothing too serious, except for the Analysis and the oncoming Algebra, so it should be a relatively easy semester.

… Which is why I agreed to TAA for another physics course …

But, new year, new semester. Clean slate. It’s invigorating, really. Indeed, in several ways things are looking -quite- up. I always feel this way, every year, and so little ever comes of it really. But I’m ok with that : ) Maybe the mere act of writing this will help spur me on to new and better things. One thing is for certain though – things are different. And so we hurtle on, into the future.

And it strikes me, I do believe, that this is as good a time as any for New Year’s Resolutions. It’s certainly no better than January 1, at least, time being so arbitrary.

Approximating Factorials: A Neat Limit

It becomes of interest, in a number of areas, to be able to approximate, for a given value of n, the value of

This is a difficult expression, as are most all expressions involving factorial, so it is convenient to have a good approximation.

Playing around with logs and integrals, I was able to derive the following approximation, that

Indeed, for large n, the two functions are practically proportional to each other. And, most interestingly of all, according to Mathematica, calculating out that limit,

Neat, no?

Generating Functions

Generating Functions are a neat tool in the study of recursive sequences.

A generating function is a clothesline on which we hang up a sequence of numbers for display.
— Herbert Wilf, Generatingfunctionology (1994)

The idea is that, given a sequence of numbers , we want to examine functions of the following form.

This is the so called ‘ordinary’ generating function. By examining properties of f(x), we can derive information about the formula for that sequence of numbers.

Recursion Grab Bag

A rather interesting topic in math (what a general way to start a post) is the matter of recursive functions. These are functions that are expressed in terms of themselves. A fairly traditional example would be factorial. n! is defined to be

Now, a nicely concise way to write this, not to mention computationally convenient, is to define the following

So, the operation is defined, effectively, with respect to itself.

There are a couple of points of interest when it comes to recursive functions. One is expressing common operations and calculations in terms of recursion – it’s often more elegant and much more natural, like the above definition of recursion. Then, there are some functions that can -only- be expressed in terms of recursion. Which is of special interest, since one of the nice things to do with recursively defined functions is to determine a formula for what that function is – and quite often, you just can’t.

It’s all very interesting.

The Necessity of Relativity

Relativity holds a rather curious place in physics, as I see it. Other branches – nuclear, mechanics, E&M, so forth, all seem firmly based in the study of physical objects, systems, and principles. Relativity is, in point of fact, much more of a math than a physics. It starts with simple axioms about our understanding of the universe and derives conclusions from them about the way the universe must work. The tools of relativity are not physical laws, but rather logic.

Now, as I said, relativity starts with simple axioms, and works from them to derive its conclusions. The axiom in question in this case is that the speed of light is finite, and constant. No matter who is measuring it, how fast they are going, what they are doing, who else is measuring, all observers always measure the same speed of light. That speed we’ll call c here, though more traditionally it is 299 792 458 meters per second.Now to the experiment.

Welcome to the Future

This is an experiment. In theory, it is the next phase of my former weblog, Foxmaths!. There are a number of reasons I’m considering this transition. In no particular order – Number one, WordPress just seems more stylish than Blogger. Number helo, WordPress allows for functionality I don’t have and wish I did with Blogger. For example, WordPress supports LaTeX, while it was merely hacked into Blogger. As a result, many math and formulae heavy posts I’ve written over the past year or so are in constant danger of vanishing into the bowels of the internet. WordPress gives me some security to that end. Number green, in this new year, I would like to take my level of math writing to a new level of polish and refinement, and the best way I think to do that is to start anew.

The point and purpose of this weblog is largely to explore certain math problems and perspectives as I see fit, basically a journal of my mathematical enterprises. While I am a mathematician by nature and by trade, on occasion I will branch further into the sciences, especially into physics and computer science. Though it’s all of a feather, isn’t it? Occasionally, I may branch more topical, politics, current events, even philosophical – though I usually leave those subjects to those more knowledgeable and more interested than I.

And so we’ll just see where this leads. Onwards, into the future.