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Not mixing is just as cool
(Mathematics Magazine, 2007)Newton's law of cooling, a staple of the Calculus curriculum, is an empirical law not meant for mathematical proof. However, we show it is mathematically equivalent to the intuitively appealing principle that the average ... 
Associativity of the Secant Method
(American Mathematical Monthly, 2002)Iterating a function like 1+1/x gives a sequence which converges to the Golden Mean but does so at a much slower rate than those sequences derived from Newton's method or the secant method. There is, however, a surprising ... 
On two types of exotic addition
(Aequationes Mathematicae, 2009)We classify, under reasonable assumptions, all differentiable functions f for which the 'secant method' [xf(y) yf(x)]/[f(y) f(x)] is continuous and associative. Further, we classify all differentiable functions for which ... 
On integral Apollonian circle packings
(Journal of Number Theory, 2006)The curvatures of four mutually tangent circles with disjoint interior form what is called a Descartes quadruple. The four smallest curvatures of circles in an Apollonian circle packing form what is called a root Descartes ...