Artefactual coattail

First a disclaimer: unless you are a person who will not be bored by this post, you almost definitely will be bored by this post, so you should probably stop reading now.

Now, forward. On my first post I posted some pseudo-math pseudo-chemistry problem that looked something like:

You have x moles of compound X, y moles of compound Y, and z moles of compound Z in a container. They react according to the following balanced chemical equations:

X + Y => XY
Y + Z => YZ
Z + X => ZX

Show that if there was no excess reagent remaining after the reaction (i.e. all of compounds X, Y, and Z were converted to XY, YZ, ZX), you can form a triangle with sides x, y, and z.

If you think about this for about five seconds, you'll realize that, although it sounds strange, it's really quite a trivial rewording of the triangle inequality. Now, the question is, can we get similar inequalities for similar systems of 'chemical' equations? I will answer this question (to the extent that I have answered it) tommorrow - right now, let's figure out what the question really is.

Firstly, chemistry is sort of related to real life, and real life has lots of little details that make life annoying. So let's establish a general system that works 'like' stochiometry, but has no dependence on it. The system will be described as thus:

We have n 'chemicals' - . We also have 'units' of chemical . Now, we define a 'rule' as an n-tuple of nonnegative real numbers, such as . At a given time, we can apply a given rule as follows:

For a positive real (this is basically 'how much' we use the rule), we can apply the rule only if (we need enough chemical to react!) for all . After we apply the rule, each decreases by .

We have 'no excess' if, after applying some combination of rules, . Otherwise, the total excess is


For example, with this new notation, our first question can be described as: For a system of chemicals, and rules show that there is only no excess if we can form a triangle with sides .

Now, onto generalizations:

Generalization 1: Let an system be a situation where there are chemicals, and we can combine any distinct chemicals in equal proportions. So, is a rule only if there are 1's and 0's. The example above is a system. The triangle inequality governs when the system has no excess. For an arbitrary $(n,k)$ system, are there any interesting i) algebraic, ii) geometric inequalities/constraints on the such that there is no excess.

Generalization 2: For any constraint found in generalization 1, if the $y_i$'s break the constraint, how much excess is there? (Either find a simple algorithm or, since the system is symmetric, make an assumption that - although, if you are persistent, you can probably find a formula just using absolute values).

Generalization 3: Let a system be a situation where there are chemicals, and we can combine any consecutive chemicals together. For example, the system has rules . It happens that the system in the example is also a system. Now, answer the same question as in generalization 1.

Generalization 4: Same as generalization 2, but this time applied to generalization 3.

Generalization 5: If there are rules and chemicals, can we say anything special about final excess? Try developing an algorithm for this prior to doing generalization 6.

Generalization 6: Can you find a general algorithm that determines the minimum excess remaining given any system of chemicals and rules? (I'll try to post my code in Java for this tomorrow).

So, we've now made ourselves an interesting (weird) linear algebra problem out of a fairly random and simple chemistry idea. If all these open and vague questions have annoyed your brain, go do these: http://www.math.uiuc.edu/~hildebr/putnam/problems/contest00problems.pdf
(they came up when I was looking for how to put LaTeX into Blogger - they're pretty fun (but unfortunately, pretty easy)).

- squid out