## Friday, May 8, 2015

### Component algebra and simple connected graphs

I wanted to go more in depth on the previous idea on the blog but I have to coin some terms to be able to write about it as I don't know what it would normally be called if it's already something that's studied. So I call it component algebra...
Each component is a variable a,b,c,... and it is related to other components through multiplication... Each somponent is equal to the multiplication of one or more of the other components, with a couple more rules that I'll show below...
So there might be a list of these component relationships like below, and a corresponding simple and connected graph, which is drawn so that components are equal to the multiplication of components that they are connected to...

And we are also interested in the solutions which can be found by variable elimination, for example plugging what one component is equal to in each of the rest of the equations and proceeding until there is only one variable in one equation, and backsolving. It is easy to see there must be always at least two solutions, every component equals 1 or every component equals 0, but in the case above there are even more...
In the above e=e is Maple's way of saying that e can be any value, so in this case there are infinitely many solutions.  So if e =2, say, the non-zero solution would be a=1/4, b=2, c=1/4, d=1/4, e=2, f=4, those we can say "satisfy" the graph... See that for instance c=1/4 = a*b*d*e = (1/4)*2*(1/4)*2=(1/4). And all the other components work as well.
The fact that the equations are related to a simple and connected graph ensures that every variable is defined, all exponents are 1, and a variable can occur on the left or right side of an equation but not both at the same time... a.k.a there are no loops in the graph. These all make the system relatively easy to solve.

The interesting thing to study is when there are more than the two trivial solutions, but I'm working on how to know when that will be... Sometimes the solutions are complex numbers as well...

#### **As sums**

Dr. Rose suggested perhaps using sums as linear equations might be simpler, for example...
Sometimes that is easier and nicer to think about but other times it misses solutions at least in Maple:

In the example above using linear sums only shows a trivial solution but the product shows more solutions...