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Series — page 2

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
One matching that is not stable, and all 24 counted by blocking pairs. An unstable matching with its blocking pair ringed and both members' rankings marked, above an exhaustive census of every matching of the instance by how many blocking pairs it has.

Stable matching

  1. 1 Nobody has a reason to run away
  2. 2 The side that proposes wins
  3. 3 No stable rule is safe from a lie
  4. 4 The people every stable answer leaves out
  5. 5 A ring that no pairing can break
  6. +6 more
11 essays · applied The circle of radius √25 on the integer lattice. A circle drawn on the whole-number grid, with the lattice points it passes through marked.

Sums of two squares

  1. 1 Two squares, and a lattice
  2. 2 The two squares actually produced
  3. 3 Almost no number is one
  4. 4 The wait for the next sum of two squares
  5. 5 A fraction on the circle forces a whole point
  6. +6 more
11 essays · number A majority cycle over 3 candidates, and how often 3 voters produce one. The majority tournament as a directed polygon with each arc's margin, beside one cell for every profile of the stated size, filled where no Condorcet winner exists.

Voting rules

  1. 1 The majority that goes in a circle
  2. 2 Five rules and five winners
  3. 3 Four conditions, and no rule that has all of them
  4. 4 A lie that pays
  5. 5 How often the majority goes in a circle
  6. +6 more
11 essays · applied Bayes' theorem as two rectangles. A unit square split by how common the condition is (1.0%) and then by how the test behaves. Of everyone who tests positive, the fraction who have it is 16.7%.

Bayes

  1. 1 Bayes' theorem is a picture of a square
  2. 2 The door that was not opened
  3. 3 Two children and the sentence about one of them
  4. 4 One coin, counted by runs and by wakings
  5. 5 Evidence measured in decibans
  6. +5 more
10 essays · probability When a shared birthday becomes likely. The chance that some pair in a group shares a birthday, against group size. It passes a half at 23 people, where the probability is 50.7%.

Birthday problem

  1. 1 Twenty-three people
  2. 2 Any unevenness brings the match sooner
  3. 3 A collision that finds a factor
  4. 4 Fourteen people within a day
  5. 5 A room where nobody is alone
  6. +5 more
10 essays · probability One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

Borsuk ulam

  1. 1 One line that halves them both
  2. 2 Two opposite points that agree twice
  3. 3 As many cuts as colours
  4. 4 Opposite labels that have to meet
  5. 5 The colours a circle forces
  6. +5 more
10 essays · topology Two polytopes, two optima, one number. The feasible regions of a linear program and of its dual, side by side, each with its optimal vertex, and a number line on which the gap between the two optima closes to nothing.

Duality

  1. 1 Two numbers that have to meet
  2. 2 What a constraint is worth
  3. 3 When one of the two numbers is missing
  4. 4 The lines the optimum lies under
  5. 5 Prices at every corner
  6. +5 more
10 essays · applied One cake, one halving cut at 4/9, and two measures of it. A cake as a bar with two step valuations above and below it, the cutter's halving cut marked, and a table of both people's exact value of each piece.

Fair division

  1. 1 One cuts and the other chooses
  2. 2 Three people and a trimmed piece
  3. 3 Envy-free, up to one item
  4. 4 How many cuts a fair share costs
  5. 5 The product that makes a division fair
  6. +5 more
10 essays · applied The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Finite geometry

  1. 1 Seven points, seven lines
  2. 2 A schedule where every pair meets once
  3. 3 More blocks than points
  4. 4 A plane in a list of numbers
  5. 5 A plane no field built
  6. +5 more
10 essays · computation Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

Harmonic series

  1. 1 A sum whose terms vanish and whose total does not
  2. 2 The same terms, in a different order, adding to whatever is asked
  3. 3 The sum that steps over every whole number
  4. 4 A coin in front of every term
  5. 5 A coin in front of every power
  6. +5 more
10 essays · analysis 3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

Judgement aggregation

  1. 1 The court that contradicts itself
  2. 2 No rule escapes the doctrinal paradox
  3. 3 Deciding the premises or the conclusion
  4. 4 Four ways out, and what each costs
  5. 5 The nearest consistent verdict
  6. +5 more
10 essays · applied Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders.

Latin squares

  1. 1 The thirty-six officers
  2. 2 A field's worth of squares
  3. 3 The plane hiding in the squares
  4. 4 Nine thousand four hundred and eight
  5. 5 Sixteen of five hundred and seventy-six
  6. +5 more
10 essays · computation the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linking number

  1. 1 Two loops and one number
  2. 2 Zero can mean two different things
  3. 3 Linked, and no two of them are
  4. 4 A whole number split into two that are not
  5. 5 Six points in space and a pair that must link
  6. +5 more
10 essays · topology 120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Monte Carlo

  1. 1 Getting pi by dropping needles on the floor
  2. 2 The error that does not care how many dimensions
  3. 3 Sampling where the answer lives
  4. 4 A walk that samples a distribution
  5. 5 Points too even to be random
  6. +5 more
10 essays · probability A set, its negation, and its double negation. Four bars on one number line showing an open set, its negation, their union, and the double negation.

Non classical logic

  1. 1 The middle that is not excluded
  2. 2 Not one step but a continuum
  3. 3 Refutable in something small
  4. 4 A proof that says which half
  5. 5 No table of truth values is enough
  6. +5 more
10 essays · logic 5,040 orders, 7 thresholds, one best rule. For each number of candidates passed over, the share of the 5,040 possible arrival orders in which the rule ends up with the best of the 7. The count is exhaustive.

Optimal stopping

  1. 1 When to stop looking
  2. 2 When the numbers are shown
  3. 3 Giving up on the best
  4. 4 Half of what an oracle takes
  5. 5 The thresholds that nest
  6. +5 more
10 essays · probability The partition 5 + 4 + 2 + 1 and its conjugate. A row of dots for each part, and the same dots read down the columns instead.

Partitions

  1. 1 A diagram turned on its side
  2. 2 Every partition, hidden in a product
  3. 3 The terms that cancel almost everything
  4. 4 The size of a number with no formula
  5. 5 Every fifth one divides
  6. +5 more
10 essays · number The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

Period-doubling

  1. 1 The road paved with doublings
  2. 2 A constant that does not care which map
  3. 3 The dark lines are one point's orbit
  4. 4 The window that opens with a stutter
  5. 5 Every window is the whole diagram again
  6. +5 more
10 essays · dynamics The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Pythagoras

  1. 1 Two squares, four triangles, and no algebra
  2. 2 Euclid proves it without moving anything
  3. 3 Every triple, on one circle
  4. 4 The rope that squares a corner
  5. 5 Two right angles and the diagonal of a box
  6. +5 more
10 essays · geometry Every order of arrival for three partners, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

Shapley value

  1. 1 The order everybody arrives in
  2. 2 None of the four conditions is spare
  3. 3 A share of the votes is not a share of the power
  4. 4 Too many orders to list
  5. 5 Sharing a cost that is not the sum of its parts
  6. +5 more
10 essays · applied

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