Introduction to Series-Parallel Duality

Series-parallel duality has been previously studied in electrical circuit theory and combinatorial theory.  The parallel sum arose naturally when resistors are connected in parallel instead of series.  Given two resistors with the positive real resistances of a and b, their combined resistance is a+b when connected in series and 1/(1/a + 1/b) when connected in parallel.  The colon notation is for the parallel sum, a:b = 1/(1/a + 1/b).  Any equation on the positive reals concerning the two sums and multiplication, can be “dualized” by applying the “take-reciprocals” map to obtain another equation.  Each number is replaced by its reciprocal and the two additions are interchanged.  The principal application considered in the essay is series-parallel duality in financial arithmetic.  The basic result is that the parallel sum of the one-shot “balloon” payments at different times that would pay off a given loan is the equal amortization payment that would pay off the loan if paid at each of those times.  By carrying through this interpretation, we see that each equation in financial arithmetic can be paired with a dual equation to reveal the structure of series-parallel duality within financial arithmetic.  The duality in economics is convex duality, and it is shown that series-parallel duality is the “derivative” of convex duality.

This paper is also a slightly expanded version of a chapter in the Intellectual Trespassing book on this website.

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