Hell Week

I’m always reluctant to call a given week my ‘hell week’, because I’m afraid some administrator somewhere will here me and think, Oh really, Mr. Fox. Let me show you what Hell Week really is.

Nevertheless, this week has been pretty wretched, from start to finish. And it hasn’t even finished yet. I’ve got a math studies test this weekend. 48-hour take-home exam.

HELL.

So, I’m going to be a little light on the math. For a bit.

But what I really want to talk about isn’t me, my troubles, my math or lack thereof. No, what I want to talk about is that damn robot in Newell-Simon hall. They have this robo-ceptionist there that answers the phone, directs calls, dispenses information, chats with you, and all kinds of cleverly robot things. He can see you and talk with you, though you have to type to him on his console. Nevertheless, very clever.

I hate him.

Because every damn time I walk by that damn robot, he says hello, or maybe a cordial “Good Evening”. And I don’t have time to stop and type to him – I’m perpetually half late to class as it is. So I keep walking. And he says after me,”… or maybe not…” You can almost hear him trailing off, forlornly.
That damn robot makes me feel guilty, every single time.

I hate him.

Puzzler #17: Cube

Puzzler, for the coming Carnival of Mathematics.

Consider if you will, a cube.

And we’re going to say that each face of the cube has a net direction. Imagine each face of the cube like a conveyor belt or a treadmill, so it moves to one side. And each direction is either up, down, left, or right, never across a diagonal, always along the edges of the face.

Now, the question is, assuming no gravity, can the directions of the faces be oriented in such a way that someone trapped on the inside, wearing a velcro suit, will be dragged around, and never get stuck in a single spot. More interestingly, can it be arranged such that the person will be dragged around like that, no matter where they start, on any face of the cube?

To make the question more specific, each face drags the person with equal force. As each corner of the room has 3 faces touching it, if at least 2 of the three faces pull the person into the corner, we’ll say that the person becomes stuck there. That’s what we want to avoid, so they just get dragged around and around, no matter where they start in the room.

So that’s the problem. Can you build such a cube. If you want to think about it before learning the answer, I’d stop reading now.

Masser-Gramain Constant

This result is probably one of the more interesting ones I’ve stumbled across in a long while.

It concerns the nature of mathematical functions. Mathematical functions are basically machines that you give a number to, and they give you another number by some rule. f(x) returning 2*x, for example. An arbitrary function though is little more than a list of numbers, and a corresponding list of numbers that they are mapped to. The basic, most complete description of that function. But, from time to time, the mapping can be described by a rule, like x goes to 2*x. x goes to 0 if x is rational or 1 if x is irrational (that’s a really interesting function). Occasionally, the rules lead to interesting properties like differentiability. Other times, the written out map is the best and only description of the function. The question is – just from the mapping, what can you tell about the function and the rules that generated it? Being able to derive information about those rules, from the mapping itself, is a very neat trick.

Grab Bag: Real and Time

“It is well known that Mahatma Gandhi’s insistence on eating his salads untossed shamed the British government into many concessions.”

It’s been a rough couple of weeks, climaxing in Friday. In no particular order, a pretty hefty math assignment involved staying up until 3+ AM, I found out I’m never going to see a friend of mine again, and there’s just been a general depression/lack of interest rampant lately. But there was a moment, walking down the street at 3 in the morning with said friend, and it had been snowing the last couple of hours, so the streets, sidewalks, and trees were covered in this powdery, undisturbed layer. The flakes continued to fall gently through the pools of light, contrasting with the pure black of the night in a way that could only be described as poetic. It was nice. For all the crap, slush, and muck Pittsburgh throws at you … it can be quite nice.

There’s something profoundly depressing to me about the writing I do here. Well, if I were to be honest with myself, many things depress me. And plenty of them have been doing a number on me lately, but I think I’m past that. I think. But let’s get down to cases.

A Summation to Consider

Consider the following.

Does that converge?

Mathematica says no. But, if instead of summing over i, you integrate over i, Mathematica says yes, which is an inherently contradictory result.

Any suggestions?

Damn Cold Outside

It’s days like this that I’m reminded of this neat little invention that came out of Stanford. It’s a box with a hole in it. You reach in the hole, and there’s a lip that forms a weak seal around your wrist. The inside of the box is kept at a lower pressure than the outside. Once you reach in, you grab hold of a chilled bar in the middle of the box. The cold chills the blood in your palm. Normally, the vessels would constrict, reducing the blood flow back to the rest of your body, but the pressure difference keeps them from doing so. As such, there’s nothing restricting the flow of the cooled blood back into your body, lowering your core temperature.

I think that’s how it works, anyway. But then again, I’m a mathematician, not a biologist. Read the article a while ago. They used it on football players, and showed there was a marked improvement in performance. Interesting stuff.

Puzzler #9: Don’t Cross the Streams

The idea is that in the plane you have m many red points and m many black points. I like to think of them as Ghosts and Ghostbusters. Significantly, the points are arrayed in general position, which is to say no three of them lie on the same line. Here’s an example for m = 5.

We’re going to pair up each black point with a red point and draw lines connecting them. For example, like so.

Now, the problem is this – that is a -bad- pairing, because some of the lines intersect, and as the title states, you don’t cross the streams. It would be bad. The question is, is there a way of pairing the points such that the lines do not intersect. In this example case, yes.

But, is it always possible? For any value of m, with the points in any general position, can they always be connected in a way that avoids intersection?

That is the puzzler. And if you want to think about it, I’d stop reading right about here.

Last warning, the answer is forthcoming….

Probabilistic Chess Colorings

Now isn’t that an inviting title.

So, the idea is that in general, you have an NxN chessboard, with all white squares, and you’re going to go through each square and color it black with a probability p. So ideally, you’re left with p*N*N many black squares and (1 – p)*N*N many white squares. Consider the 20×20 board below. I hope this comes out formatted properly.

X__X_XXXX________XX____X_X__XX
_X_____XXX____X_X_XXX___X_X__X
_______X___X_XX_____XXXX_X__X_
X__XX____XXX__X_X__X_____X_X__
X___XX__X__XXXXXX____XXX__X__X
___________XXXXX__XX__X_XX__XX
XX__XX___X__X___X___X_____X__X
X_X_X_XXX_______X___X_____XX_X
_XXX__X_X___XX_X__XXX__X___XX_
X___X__X_XX_X_XXX____X_____X_X
X_X_X_X____X_X_______________X
_XXXX_X__X_____XX_X_X___X___X_
____XX_XX_X___XX____XX_X____X_
X_X_XXXXXX____XX_X_____X____X_
X___X_X_XX______XX_X____XX_XXX
X_X__X__X___XX_XXX____X___XXX_
_X_X_XX_X____X_______XXX__XX_X
_X_XXX_______XX__XXX__XX__X_XX
____X_______X_XXX_XX_X_X______
__XX_X__XX_X_X___XX_XX_X__XX_X
X___XX_XXXXX__X____XXX_X______
__X_XX__XX_X__________XX__XX_X
______XXXXX_______X_X________X
X__X_X__XX__XX_XX_XX_____XXXX_
X___XX_XXXXX_XXX_X_XX___XXX_X_
_XX_____X_XX_X___X_X___XXXXXX_
__XX_X_X____X_X__X_______X_X__
X__X_X___X_XXX_XXX_X__XX_X__X_
___X_X__X_XXX_____XX__XX______
_X_XX_X__X___XXX_X__X_X_______

The question I would like to consider then is, what is the size, in number of squares, of the largest continuous black section? We’ll say that two black squares are touching only if they share a side – no diagonals. So, for a given value of p on a given board, what is the largest number of black squares you expect to find all touching each other?

This is a variant of a problem in graph theory known as looking for the largest connected component of a random graph. Analysis of that problem yields some interesting results, but I’m not sure how to extend it to here, where the geometry of the graph is modified.

However, we can approximate, test, and simulate. Using p from 0 to 1, at increments of 0.01, I generated such boards (I think it was a 70×70 board), filled them, and determined the maximum connected component on the board. And then I repeated this multiple times, averaging them all together with a convenient ruby script. Here is a plot of the results, in terms of the probability p along the x, and the maximum size as a fraction of the whole board along the y. Click for the full graph. Won’t fit nicely on the main page.

Notice that for p less than about 1/2, the size of the maximum component is ridiculously small. Which means that if you color only half or less of the squares, you can expect, on average, very tiny connected blobs of black. But soon after that, the behavior of the curve changes drastically, practically exponentially. A dramatic increase in the expected size of the maximum component. It is as though the board reaches a sort of carrying capacity for the smaller blobs, after which point (increasing p and adding more black squares), it becomes necessary to begin connecting smaller blobs into larger. There’s no way to fit any more smaller blobs onto the board, and as you continue to add black squares, the smaller blobs become connected at an increasing rate.

It’s very interesting, I think.

And makes me think, abstractly, of Blokus.

Headaches: Part 1 of a Continuing Saga

It strikes me that headaches aren’t really like any other kind of pain. Stomach, arm, leg … you could break your arm and muddle through. You could get shot in the shoulder and take yourself to the hospital. There’s … a distance. Between you and the pain. (This doesn’t necessarily hold for being stung on the eyeball by a Bullet Ant). But headaches are a different story. They crawl inside of you and possess you. They’re in your mind and in your thoughts, gnawing at your very soul.

Headaches are singular.

My headaches are usually very localized, and this morning was no different. A sharp pain, right through my left eyeball. I lay there in bed for about half an hour, trying to figure out the form of it. The shape. For a while, I thought it just shot straight through my eye to the back of my head. But if I concentrated, I could feel the pain glancing downwards, into the base of my skull. And at the same time curving upwards into the top. It was like my initial perception was simply the average of those two sensations. Ultimately I was able to drag myself out of bed. After a dose of caffeine and pills, about an hour later the pain migrated, curling up in my right temple, which made me able to see finally without pain. Always a good thing. Things are mostly alright now, but for the faintest of pressure differences from one side of my face to the other, and a keen awareness of how my teeth don’t fit together properly. An exciting morning.

Math problem. We’re interested in the integral solutions to the equation

Clearly there is the trivial solution, x = 0, y = 0, z = 0. But are there any more?

Complex Numbers Make Life Easy

Consider the following integral.

They love to ask this integral in basic calc courses, and that sort of thing. And there are two very different ways of evaluating it that you could take, one, in my opinion, vastly superior to the other.