Essays
These are our essays. They showcase applied mathematics across a wide span of subjects, from physics and engineering to finance, history and medicine: the models, the methods that solve them, the code that runs them and the algorithms behind them, often with a live model you can try for yourself. Each essay starts from a question and works it through to a number. Nothing here is a product or advice. The code we publish as Jupyter notebooks is under Notebooks, the live boards that each answer one practical question are under Tools, and the research papers are under Research Papers. Every essay belongs to one of our five research desks; the buttons below show one desk's essays at a time.
1. The Woman Who Outlived Her Buyer
In 1965 a lawyer in Arles bought the flat of Jeanne Calment, 90, for 2,500 francs a month until she died. The life table expected a bill of about 144,600 francs. He died thirty years later, having paid about 900,000. Gompertz's law puts her 122 years at 1 in 81 billion or worse; the plateau measured in Italians over 105, at 1 in 15 million.
Read more →2. Missing the 10 Best Days
Fund companies print it every year: miss the 10 best days and your money is cut in half. On the S&P 500 with dividends since 1990, a dollar became 44.6, or 19.8 without its 10 best days. Miss the 10 worst too and it became 47.6, more than staying in. And 7 of the 10 best days came within a month of one of the worst.
Read more →3. Four Hundred Dollars or Nine Cents
A dollar in TQQQ, the 3x Nasdaq-100 fund, became about 408 dollars between 2010 and 2026 while the plain fund made 20. The same daily-reset machine run on the S&P 500 from 1998 to 2009 turned a dollar into 9 cents while the index, with dividends, went nowhere. One line of Itô calculus explains both: a leveraged fund is a power of the index taxed by its variance, and it wins only when its leverage is below the market's best, (μ − r)/σ².
Read more →4. Reject Three Out of Four
The random walk that estimates central-bank models, dated the start of the HIV pandemic and measured the age of the universe works best when it rejects about three proposals in four. In many dimensions the walk becomes a Langevin diffusion, and 23.4% is where that diffusion runs fastest; a walk that accepts 90% needs 23 times the steps.
Read more →5. Right 50.75% of the Time
Medallion, the fund that made 66% a year, was reportedly right on just 50.75% of its trades, and it placed 150,000 to 300,000 of them a day. The arithmetic shows how that tiny edge becomes a fortune: at least 4,505 independent bets a year, a correlation between them below 1 in 4,504, and a stake that sets the chance of ever losing half the money whatever the edge: 50% at full Kelly, 12.5% at half.
Read more →6. What a Stop-Loss Costs
In a random walk every stop-loss lowers what you can expect to end with. On the S&P 500 since 1990 a 5% trailing stop cost a quarter of the money, while a 10% one roughly broke even and cut the worst fall from 55% to 34%: the width of the stop decided everything. For a price that comes back, a stop has a stranger effect: it puts a floor under the price you should buy at, and set too tight, it kills the trade.
Read more →7. Should You Wait for a Dip?
Waiting for a 10% dip before buying the S&P 500 left the waiter behind in 72% of the months since 1990, and a short theorem says why: when the market beats cash, every waiting rule loses on average. For a price that comes back, waiting is right. The textbook answer for one trade says wait for a fall of more than two typical swings; a trader who keeps trading should buy at less than one.
Read more →8. The Shock That Refuses to Break
A weak tidal bore does not arrive as a wall but as a train of waves, and its first wave grows to twice the height of the step. The Korteweg–de Vries equation explains both, and one more thing: the energy a breaking bore burns, an undular bore keeps, at exactly the same rate, in a wave train that never stops growing.
Read more →9. The 1966 Retiree
A retiree of 1966 who drew 4% a year from the stock market ran out of money in 1992, though the market earned 4.87% a year after inflation; the same thirty years in reverse order would have left 2.34 times the starting sum. Across 126 retirements since 1871, 4% failed three times and 5% failed 24 times, and Dufresne's identity gives the chance that a withdrawal ever runs out in closed form.
Read more →10. The Rule That Fits One Ward
Hospitals are told to keep their beds at most 85% full. At that occupancy a 100-bed ward is full about once a month, a 20-bed ward within a week and a 500-bed hospital about once in six years. The Ornstein–Uhlenbeck process explains why: what matters is the distance to a full ward in standard deviations, and it grows with the square root of the size.
Read more →11. The Barrier That Comes Early
A price starts at $100 and the average wait until it first reaches $120 is nine years. Yet there is a 40% chance it gets there within a year, and the most likely moment is three months away. Under geometric Brownian motion the whole distribution of that wait is one closed formula, which prices an up-and-out call and a clause in a project contract alike.
Read more →12. The Birds That Turn as One
A starling watches its six or seven nearest neighbours, however far away they are. Put that rule and its obvious rival into Vicsek's flock of 400 birds and the rival wins when the birds are packed close, then collapses when the flock spreads out. Counting costs a little agreement and buys a flock that does not break.
Read more →13. The IQ Map That Filled Its Own Blanks
The world IQ map gives 200 countries a colour, and they all look equally sure. Behind the United States stand 62,296 people; behind Angola, nineteen; behind fifty countries, nobody, their colour copied from next door, and five sit at a floor of 60 that no test produced. Yet the measured part holds up: it tracks the World Bank's school tests at r = 0.81. The copies track nothing.
Read more →14. The Wave That Nobody Ordered
In the Beer Game, customer demand steps once from 4 cases a week to 8, yet the factory's orders swing to 33. Written as 27 differential equations and fed the ordering rule Sterman fitted to real players, the model reproduces their waves, and shows the one number that stops them: what was already on its way.
Read more →15. The Cell That Solves Bellman’s Equation
A bacterium shifted to better food must decide, protein by protein, whether to build ribosomes or the enzymes that feed them. Posed as optimal control, Bellman's equation says the answer is all or nothing, switched along a single curve. E. coli has no brain, but its alarm molecule ppGpp switches the same way, and gets 98% of what is possible.
Read more →16. The Equation That Paints a Jaguar
In 1952 Alan Turing wrote down two chemicals and a race, and showed that spots can appear from nothing. Seventy years on, a pair of his equations paints a jaguar: a spotted cub first, then broken rosettes with dark centres, from three changes of rate. And in fetal house-cat skin, biologists have now caught the real thing: an activator and an inhibitor laying down the stripes before birth.
Read more →17. The Best of 490 Backtests Is Noise
Search a few hundred strategy variants on one year of data and the best one scores a Sharpe near three — with no edge in any of them. The arithmetic, the figure, and what survives it.
Read more →18. The Battleship That Solved a Differential Equation
The 16-inch guns of USS Iowa threw a 1,225 kg shell 38.7 km, and aiming them is a boundary-value problem for an ordinary differential equation. Fitted to the Navy's range tables, the model finds the gun's best angle beyond its turret's 45-degree stop, a bunker that breaks only from farther away, and a weaker charge that spares a town.
Read more →19. The Head That Was Written in Harmonics
Marcus Agrippa had the Roman empire surveyed and its map engraved on marble. This post maps his head onto a globe: a scanned portrait relaxed conformally onto the sphere, then written as a series of spherical harmonics, 786,432 numbers to degree 511. The series closes in on the stone as the degree rises, and the last place it arrives is the sharpest edge on the head.
Read more →20. The Motor That Counted Its Own Steps
Kinesin walks along the rails of a nerve cell 8 nm at a time, and only the steps can be seen. In 1994 the noise of those steps showed that two or more hidden processes drive each one. The theorem behind that inference, proved by Aldous and Shepp in 1987, says no cycle of n hidden states can keep better time than n equal stages in a line.
Read more →21. The Third Walker
Einstein and Gödel walked home together for years, and on one question they never settled: Gödel's rotating universe, where a traveller can loop back into the past, and his claim that it proves the passing of time is unreal, here too. Einstein called it important and left it open. Hermann Weyl worked down the corridor. This is the walk where he joins them.
Read more →22. The Catch That Emptied the Grand Banks
In 1992 Canada closed the northern cod fishery, its stock at about 1% of what it had been. A fixed quota at the maximum sustainable yield sounds prudent, but its balance point is a cliff edge: with good and bad years, 93% of model stocks fished that way collapse within 50 years. The optimal rule, solved exactly, takes more in good years, nothing in bad ones, and never collapses from fishing.
Read more →23. The Cable That Whispered Too Slowly
In 1858 the first Atlantic cable carried about a tenth of a word a minute. William Thomson had predicted why in 1855: a long cable carries a signal the way a bar carries heat, and its delay grows with the square of its length. The chief electrician called that "a fiction of the schools" and pushed up to 2,000 volts down the cable. The equation was linear; the volts changed no time, and within weeks the cable was dead.
Read more →24. The Lease That Was as Good as Forever
In 1898 Britain leased Hong Kong's New Territories for 99 years, a term its negotiator thought "as good as forever". In money he was right: at 5% a year, 99 years hold 99.2% of forever. A slow walk from a project's value forever to its value for T years, first by subtracting forever from forever, then by making the end random, which turns out to be compound interest run backwards.
Read more →25. The Fleet That Lost by Squaring
At Trafalgar 27 British ships of the line met 33. Lanchester's square law says that is 729 against 1,089: fought line against line, the British fleet is gone and 19 allied ships remain. Nelson's memorandum, written 111 years before the law, does its arithmetic, and solving the equations shows where the advantage really comes from: the enemy's idle guns.
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