Concept

Linear system

A collection of linear equations in several unknowns, solved by elimination. A great many quantities in probability that look like limits — expected times, absorption chances, hitting times — are exact solutions of one instead.

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

How many colourings each knot allows. Three knots, and the number of ways their arcs can be coloured with three, five and seven colours under the crossing rule, beside the determinant computed separately from the same crossings.

Colours that count more than three

Three colours prove the trefoil is knotted and say nothing at all about the figure-eight, which refuses them exactly as an unknotted loop does. The repair is to stop colouring and start counting — with five colours, or seven, and with the arithmetic done modulo the number of them.

topology · Knots

A game that stops, over totals 0 to 5. States in a row with arrows up and down between them and the two ends absorbing, above a table of the expected number of steps and the chance of ending at the top from each start.

The chain that stops

Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.

probability · Markov chains

Named alongside it

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

Invariant Absorbing state Counting argument Decidability Expectation Knot Knot determinant Markov chain Matrix Modular arithmetic Probability Random walk

All concepts