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Concept

Uniqueness

The property of a specification that exactly one object satisfies it. A characterisation needs two halves — that the conditions leave only one object, and that each condition is needed, which is shown by exhibiting something satisfying all the others and failing that one.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.

analysis · The exponential Drop one condition, and something else satisfies the rest. A column for each of the four conditions, holding a sharing rule that breaks that one and keeps the other three, with the split each rule gives on a stated four-player game.

None of the four conditions is spare

Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.

applied · Shapley value The numbers to 18 in hereditary base 2, and their ordinals. A table of small whole numbers written in hereditary base notation beside the ordinal obtained by replacing the base with omega.

Every ordinal in base omega

Every ordinal below a certain point is a descending sum of powers of ω, in exactly one way. That notation makes comparison mechanical, it is what hereditary base notation becomes when the base is replaced, and it stops at the first ordinal it cannot name.

logic · Ordinals A parallelepiped of volume 2.94. The image of the unit cube under a three-by-three matrix, beside the six signed products whose sum is its volume.

The only function that behaves like a volume

Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.

algebra · Determinant How many codewords lie within each radius, for a [7,3] code over 11 symbols. A bar for each decoding radius, its height the largest number of codewords found inside a ball of that radius around a randomly drawn received word, with the unique-decoding radius and the Johnson radius marked.

Past half the distance

A code of minimum distance five corrects two errors, and every account stops there. Two is the largest number for which the answer is unique — and a decoder that returns a short list instead of one answer reaches considerably further, which can be measured by counting the codewords in a ball.

computation · Error-correcting codes Shares on two triangles and a go-between. A network of players with each node labelled by its share of what the whole network earns, and its number of links beneath it.

Cutting a link costs both of its ends the same

Three players, any two of whom can earn 1 together — but only if they are linked. Link all three and each is due a third. Remove one link and the player holding both of the others is due two thirds. Averaging over orders on the game the network allows is the one rule under which breaking any link costs the two players it joined exactly the same, and it pays go-betweens more than their links.

applied · Shapley value The factorials, and three smooth curves through all of them. Γ(x+1), Γ(x+1)(1 + 0.5 sin² πx) and the degree-six interpolating polynomial through 0!…6!, on a log scale over 0 ≤ x ≤ 5.4; at x = 1/2: 0.88623, 1.32934, -3.58301.

Only one curve through the factorials bends the right way

Infinitely many smooth curves pass through 1, 1, 2, 6, 24, 120, and some of them even obey the factorial's own rule, f(x + 1) = (x + 1)·f(x), at every x. Ask that the logarithm of the curve bend upwards everywhere and exactly one survives — Euler's integral. A wiggle of any size breaks the condition somewhere, the smaller the wiggle the further out, and the proof that nothing else survives is a squeeze that computes √π along the way.

analysis · The integral

Named alongside it

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

Cooperative gameIntegralMarginal contributionShapley valueSymmetryAdditivityAxiomAxiomatic characterisationBasisBoundConnected componentConvexity

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