Then I thought that theres almost no real use for it.

In the end I can see some limited use, but not to justify the amount of effort taken to do it. Sorry, hang on, you need to watch this first.

or watch it on vimeo.

Ok, so the gist of it is that you can use a few well placed high quality photographs to increase the resolution and dynamic range of low quality videos. Sounds great right?

Well, yes, but only when you think about the cases it uses. Basically tracking shots of static things, taken in low quality video, like the kind of video an average digital camera takes. So you then take a few still shots of the scene to boost the resolution.

Great, but just why would I do that? Almost the whole reason for taking quick videos like that is not to show a simple scene, but some action that you can’t get with a still shot.

So in review, I would give it about a B-, good effort, but honestly come on guys, not really that useful, though I don’t doubt some of the maths would be very interesting.

More on it here.

So what are friendly numbers? Lets do this quick.

We need first to get define a divisor function over the integers, written σ(n) if you’re so inclined. To get it first we get all the integers that divide into n. So for 3, it’s 1 and 3. For 4, it’s 1, 2, and 4, and for 5 it’s only 1 and 5.

Now sum them to get σ(n). So σ(3) = 1 + 3 = 4, or σ(4) = 1 + 2 + 4 = 6, and so on.

For each of these n, there is something called a characteristic ratio. Now that’s just the divors function over the integer itself ( σ(n)/n . So the characteristic ratio where n = 6 is σ(6)/6 = 12/6 =2.

Once you have the characteristic ratio for any integer n, any other integers that share the same chacteristic are called friendly with each other. So to put it simply a friendly number is any integer that shares its characteristic ratio with at least one other integer. The converse of that is called a solitary number, where it doesn’t share it’s characteristic with anyone else.

1,2,3,4 and 5 are solitary. 6 is friendly with 28. ( σ(6)/6 = (1+2+3+6)/6 = 12/6 = 2 = 56/28 = (1+2+4+7+14+28)/28 = σ(28)/28.

You may be now be thinking the following:

1. That’s all well and good, but what are they good for?
2. Who cares?

So what are they good for?

I dunno. But it’s interesting no? Actually while on the face of it, it all seems quite straight forward it quickly becomes far more complex than you would think.

Some numbers are easily proved to be solitary. Other’s we know to be friendly. But my-o-my there are alot more we don’t know. I mean look. Here is the complete list of friendly numbers we know about. And I mean COMPLETE.

6, 12, 24, 28, 30, 40, 42, 56, 60, 66, 78, 80, 84, 96, 102, 108, 114, 120, 132, 135, 138, 140, 150, 168, 174, 186, 200, 204, 210, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 294, 300, 308, 312, 318, 330, 348, 354, 360, 364, 366, 372

That’s it. What about 10, 14, 15, or 20? Don’t know. Infact it’s on the list of unsolved Mathematical problems at wolfram. Doubt there’s a cash prize involved but why not give it a go. It’s harder that it looks.

Wow.
Well done. You read all the way to the end. To give you a treat I present this lovely video. It’s a visualization of Lovely Head by Goldfrapp. Yes it’s got lot’s of maths in. No it doesn’t have anything to do with friendly numbers.

Click here to view the embedded video.
Or even better, watch it in hd.

If you were wondering. No. I didn’t just get this all off the top of my head. You can read more about friendly numbers at wolfram, or wikipedia, and here is where the full list of friendly numbers came from. Personally I find amicable pairs more interesting but then, who really cares about that?

Found here, on Flickr.

Isn’t that awesome?

You know it is.

(on a side not, I actually missed it by 2 days, so if we’re being honest it’s more like 24/7, or 3.4285… – but thats not as catchy is it?)

0. Newtonian gravity is your high-school girlfriend. As your first encounter with physics, she’s amazing. You will never forget Newtonian gravity, even if you’re not in touch very much anymore.

1. Electrodynamics is your college girlfriend. Pretty complex, you probably won’t date long enough to really understand her.

2. Special relativity is the girl you meet at the dorm party while you’re dating electrodynamics. You make out. It’s not really cheating because it’s not like you call her back. But you have a sneaking suspicion she knows electrodynamics and told her everything.

3. Quantum mechanics is the girl you meet at the poetry reading. Everyone thinks she’s really interesting and people you don’t know are obsessed about her. You go out. It turns out that she’s pretty complicated and has some issues. Later, after you’ve broken up, you wonder if her aura of mystery is actually just confusion.

4. General relativity is your high-school girlfriend all grown up. Man, she is amazing. You sort of regret not keeping in touch. She hates quantum mechanics for obscure reasons.

5. Quantum field theory is from overseas, but she doesn’t really have an accent. You fall deeply in love, but she treats you horribly. You are pretty sure she’s fooling around with half of your friends, but you don’t care. You know it will end badly.

6. Cosmology is the girl that doesn’t really date, but has lots of hot friends. Some people date cosmology just to hang out with her friends.

7. Analytical classical mechanics is a bit older, and knows stuff you don’t.

8. String theory is off in her own little world. She is either profound or insane. If you start dating, you never see your friends anymore. It’s just string theory, 24/7.

The cocky exponential function e^x is strolling along the road insulting the functions he sees walking by. He scoffs at a wandering polynomial for the shortness of its Taylor series. He snickers at a passing smooth function of compact support and its glaring lack of a convergent power series about many of its points. He positively laughs as he passes |x| for being nondifferentiable at the origin. He smiles, thinking to himself, “Damn, it’s great to be e^x. I’m real analytic everywhere. I’m my own derivative. I blow up faster than anybody and shrink faster too. All the other functions suck.”

Lost in his own egomania, he collides with the constant function 3, who is running in terror in the opposite direction.

“What’s wrong with you? Why don’t you look where you’re going?” demands e^x. He then sees the fear in 3’s eyes and says “You look terrified!”

“I am!” says the panicky 3. “There’s a differential operator just around the corner. If he differentiates me, I’ll be reduced to nothing! I’ve got to get away!” With that, 3 continues to dash off.

“Stupid constant,” thinks e^x. “I’ve got nothing to fear from a differential operator. He can keep differentiating me as long as he wants, and I’ll still be there.”

So he scouts off to find the operator and gloat in his smooth glory. He rounds the corner and defiantly introduces himself to the operator. “Hi. I’m e^x.”

“Hi. I’m d / dy.”

Enter maths.

Lets make some initial conditions and normalize our temperature scale to room temp., ie. 0 degrees = room temperature.

Now assuming this is an ordinary mug the coffee is in, nothing special will happen in the cooling. Thus we can assume that the coffee will cool at a proportional rate to the temperature difference between it and the room temp. Further to that, the amount of milk added is small enough to not affect that rate.

Some quick calculus will show how the coffee temperature decays exponentially over time, ie.

also,

We can assume that the difference between the specific heats of the coffee and milk are negligle, hence if we add milk at temperature M, to coffee at temperature C, the resulting mix has a temperature of aM+bC, where a and b are constants between 0 and 1, with a+b=1. (ie. the a and b are the relative volumes of milk and coffee of the final volume)

So, lets assign some variables.

We can denote the starting coffee temperature by C, and the starting milk temperature by M. Hence -

Thus, the difference is d=(1-l)aM. As l<1 and a>0, so now we need to worry about whether M is positive or not.

Case 1. Warm milk – you should add the milk just before you are to drink the coffee.

Case 2. Room Temperature Milk – It really doesn’t matter when you add the milk. Do it now, do it later, I really don’t care.

Case 3. Cold milk – its best add this right when the coffee gets to you.

To figure all this out without even touching any of the maths all you need to do (as with so many things in maths) is to consider the extreme examples.

For instance, lets assume you’ve got a coffee at room temperature and the milk you are to add is either really hot or just above freezing. So it becomes obvious that you should add the hot milk later, the cold milk early.

Further variations

For this entire problem we have assumed that the milk’s temperature is constant throughout, up until you add it to to the coffee. What happens if this isn’t the case? ie. you can let the milk stand at room temperature.

For this, let r = the exponential decay constant for the milk’s container.

So now we can add the acclimated milk later, giving -

This gives us a whole slew of new cases.

r<l: The milk pot is larger than your coffee cup.
(E.g, it really is a pot.)
r>l: The milk pot is smaller than your coffee cup.
(E.g., it’s one of those tiny single-serving things.)
M>0: The milk is warm.
M<0: The milk is cold.

If you’re interested in the derivation you must be a really sad individual, so lets just jump to the end and the conclusions:

Add warm milk in large pots LATER.
Add warm milk in small pots NOW.
Add cold milk in large pots NOW.
Add cold milk in small pots LATER.

Of course, observe that the above summary holds for the case where the
milk pot is allowed to acclimate; just treat the pot as of infinite
size and the problem goes away. Marvelous.

Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the grounds that it was insufficient and made her way in amongst the complex elements.

Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Quite sudenly, 3 branches of a hyperbola touched het at a single point. She oscillated violently, lost all sense of directrix, and went completely divergent. As she reached a turning point, she tripped over a square root protruding from the erf and plunged headlong down a steep gradient.

When she was differentiated once more, she found herself, apparently alone, in a non-Euclidean space. She was being watched, however. That smooth operator, Curly Pi, was lurking inner product.

As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. Was she still convergent, he wondered. He decided to integrate improperly at once. Hearing a vulgar fraction behind her, Polly turned around and saw Curly Pi approaching with his power series extrapolated. She could see at once, by his degenerate conic and his dissipated terms, that he was up to no good.

“Eureka,” she gasped.

“Ho, ho,” he said.

“What a symmetric little polynomial you are. I can see you are bubbling over with secs.”

“Oh, sir,” she protested. “Keep away from me. I haven’t got my brackets on.”

“Calm yourself, my dear,” said our suave operator. “Your fears are purely imaginary.”

“I, I,” she thought, “perhaps he’s homogeneous then.”

“What order are you?” the brute demanded.

“Seventeen,” replied Polly. Curly leered.

“I suppose you’ve never been operated on yet?” he asked.

“Of course not!” Polly cried indignantly. “I’m absolutely convergent.”

“Come, come,” said Curly, “let’s off to a decimal place I know and I’ll take you to the limit.”

“Never,” gasped Polly. “Exchlf,” he swore, using the vilest oath he knew.

His patience was gone. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly. All was up. She felt his hand tending to her asymptotic limit. Her convergence would soon be gone forever. There was no mercy, for Curly was a heavyside operator. He integrated by parts. He integrated by partial fractions. The complex beast even went all the way around and did a counter integration. What an indignity to be multiply connected on her first integration. Curly went on operating until he was absolutely and completely orthogonal.

When Polly got home that night, her mother noticed that she was no longer piecewise continuous, but had been truncated in several places. But it was too late to differentiate now. As the months went by, Polly’s denominator increased monotonically. Finally, she went to L’Hopital and generated a small but pathological function which left surds all over the place and drove Polly to deviation.

The moral of our sad story is this:

If you want to keep your expression convergent, never allow them a single degree of freedom.

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.

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