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mathematics, in pictures
The collection

Every essay — page 35

Page 35 of 35, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Applied

A rule for choosing, stated exactly, and what it forces on whoever adopts it.

Two triangles and three paths: the relaxation against the shortest tour. k = 3: 12 cities; subtour relaxation 12 with 6 half-edges; shortest tour 14; ratio 1.167.

A bound that may be off by a third

5 figures Christofides' algorithm: tree, pairing, tour. 12 cities: tree 2.4441, 6 odd cities paired for 1.5306, tour 3.6069, shortest 3.4606.

Tours within half again of the best

5 figures What survives on each agenda, 3 judges. A table of four agendas with, for each, the size of its largest inconsistent set, the number of independent unanimous rules for 3 judges, how many are consistent on every profile, and how many of those are dictatorships, oligarchies and other rules: two unconnected questions 324 of 324; a chain of two thresholds 129 of 324; two premises and their conjunction 7 of 5,832; a ranking of three options 3 of 5,832.

What the agenda leaves standing

7 figures · new Three splits that between them beat every other. The triangle of splits of one pound among three players, with the three half-and-half splits marked and every other lattice point coloured by which of the three dominates it: 134, 134, 134 beaten by exactly one, 91 by more.

Three splits that answer every other

6 figures · new