An Arithmetic and Geometric Mean Invariant
arXiv:1203.4617h-index: 4
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Provides a theoretical invariance result for interval partitioning, but is incremental for mathematics.
The paper proves that partitioning a positive real interval into sub-intervals with constant ratio of width to average value yields the same log scale for both arithmetic and geometric means, and extends this to the continuous case.
A positive real interval, [a, b], can be partitioned into sub-intervals such that sub-interval widths divided by sub-interval "average" values remains constant. That both Arithmetic Mean and Geometric Mean "average" values produce constant ratios for the same log scale is the stated invariance proved in this short note. The continuous analog is briefly considered and shown to have similar properties.