Matpic
mathematics, in pictures
Concept

Exponential decay

A quantity decays exponentially when each step multiplies it by the same factor below one. On a logarithmic scale it falls along a straight line, and the slope is the logarithm of that factor, which is how such rates are usually read from data.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A square profile of heat, spreading. The same profile at four times, each drawn from the same harmonics with each one damped by the exponential of minus its frequency squared times the time. The corners go first.

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

analysis · Fourier series Root two, raised to itself for ever. The cobweb of x to the power of root two from 1: 1.00000, 1.41421, 1.63253, 1.76084, 1.84091, 1.89271, 1.92700, settling on 2; the other solution 4 repels, with slope ln 4.

Root two, raised to itself for ever

Stack √2 to the power √2 to the power √2 without end and the tower settles at exactly 2. Solve the equation for a tower equal to 4 and √2 appears again, and that answer is false. The bases that give a limit run from e^(−e) to e^(1/e), the slope at the crossing decides everything, and a sum over labelled trees turns out to be the same number.

analysis · The exponential

Named alongside it

The objects these essays reach for when they reach for this one.

ConvergenceConvergence rateDiffusionEigenfunctionFixed pointFourier analysisHeat equationIrreversibilityIterationLabelled treePeriod-doublingPower series

All concepts