Archive for mathematics
January 3, 2007 at 8:25 am · Filed under mathematics, random
I don’t own a horse. I have only ever ridden on one when I was younger, closer to the age of 6 or 7. The horse I rode on was white. Being a mathematician I can prove this to you. So lets go:
Lemma 1. All horses are the same color
Proof.
Clearly one horse is the same color. Thus, lets assume the proposition P(k) that k horses are the same color. By induction we can imply that k+1 horses are the same color. From a set of k+1 horses we can remove one horse, and the remaining k horses will be the same color, by hypothesis.
We remove another horse and replace the first, thus the k horses are the same color, again by hypothesis. By induction we can repeat for the whole set of k horses, thus it follows that all k horses are the same color, asP(k) => P(k+1).
As P(1) is true, by initial conditions (the horse I rode as a child), P holds true for all subsequent values of k. []
Theorem 1. Every horse has an infinite number of legs.
Proof.
The horse I rode as a child had an even number of legs. Of the legs, there were two front legs, with fore legs remaining. That gave it 6 legs, which is an odd number of legs for a horse. The only number that is both odd and even is infinity. Thus, my horse had an infinite number of legs.
To show that all horses have an infinite number of legs, lets assume the inverse. Thus there is a horse somewhere with a finite number of legs. By example, this horse is pictured below:
Clearly this horse does not have an infinite number of legs. But this horse is brown, not white in color, and so, byLemma 1, does not exist. =><= []
Corollary 1. Everything is the same color.
Proof.
Lemma 1 can be generalized as the proof by induction is object independent. Hence, the statement “For all x, if x is a horse, x will be the same color”, can be generalized to “For all x, x will be the same color”. Thus proof of the single color of horses is simply a special case of this. []
Corollary 2. Everything is white.
Proof.
If a sentential formula in x is logically valid, then any particular substitution instance of it, is also a true statement. By personal experience it is evident that white horses exist. Therefore all horses are white. Hence, by corollary 1, everything is white. []
So there you have it. Clear, reasoned proof that all horses have an infinite number of legs and that everything is white.
December 18, 2006 at 7:09 am · Filed under mathematics, music, random
(of order two)
If I were somehow forced to write a song to woo a young math lady this is what it would be like.
Yes I know I’m a math nerd, but did YOU get all the maths jokes.
In case you really want to know, the lyrics are here too…
The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true
But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two
I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way
Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two
Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified
When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense
I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two
I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")
I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.
Its possibly the most awesome math based love song I’ve hear all morning.
If you want, you can even download it. Isn’t that nice.
Finite Simple Group (or order two) - The Klein Four Group
October 26, 2006 at 9:48 am · Filed under code, mathematics, projects
A while ago I completed my Mathematics BSc from Bangor University. Its a course they’ve stopped doing now, but at the time, it was a most excellent thing. For those that are interested I did my final year dissertation on Flocking Autonomous Agents. That is, the order that appears when groups of supposedly simple agents congregate.
Its the sort of thing that you see in the wild with swallows flying about. Theres no master plan of where they’re going, nor any central command to the system. So how do these simple creatures manage to form cohesive flocks that can travel, with what seems, singular purpose?
It was just this question I proposed to answer in my dissertation (and I thought I did that rather well) as well as explaining the deceptively complex maths behind it.
Long story short. The system works like this:
- The individual agents are only aware of whats going on in an area directly around them (their vision range).
- The individual agents will try to adjust their direction to that of their neighbors.
- They won’t always do this right.
- There is 4th rule.
It seems almost counter-intuitive that the fact that a agent makes mistakes in their headings would actually help the flock as a whole but, after some rather intense mathematics, you can prove that, up to a point, it helps the system.
The crux of the matter is this - If there is group of agents flying together in sync and none of them ever makes a mistake as to their heading or position, their relative positions will conceivably never change.
If however they do make mistakes something wonderful happens.
Mistakes that have a negative impact on the flock as a whole get naturally canceled out while mistakes that bring the flock together are reinforced. Over time a very tight, cohesive, resilient flock is formed.
I even wrote a lovely java applet for it and ill post it up here when i figure out how to embed it properly. Ill even give you the source code, cos I’m nice like that.
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