Applied Mathematics
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 →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 →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 →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 →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 →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 →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 →When to Shoot
Two duellists walk towards each other, one bullet each. Fire too soon and you will probably miss, and a miss is fatal. In 1949 David Blackwell worked out the answer in a day: fire when your two chances add up to one, the good shot and the bad shot at the same instant. Then he made the pistols silent, and the only safe strategy became a roll of the dice.
Read more →The Game That Remembered
Roulette cannot be beaten because the wheel forgets every spin. A deck of cards does not forget. In 1961 a young mathematician proved it, took the proof to Reno and won $11,000 in a weekend. Then he found the harder truth: with a real edge you can still go broke, and betting twice the right amount earns you exactly nothing.
Read more →The System That Could Not Lose
Dostoevsky wrote home from Wiesbaden that he had found the secret of roulette: you just can't lose. Then he signed away his future, gambled the advance, and dictated a novel in 26 days to save himself. The doubling system every gambler knew wins 39 nights out of 40 — and loses exactly the house edge anyway.
Read more →The Tanks That Counted Themselves
Before D-Day the Allies needed to know how many tanks Germany built each month. The spies said about 1,400. A few statisticians read the serial numbers off wrecked gearboxes and said 246. The records, opened after the war, said 245 — and the best estimate throws away every serial number but one.
Read more →The Forest That Burns All at Once
Over forty fire seasons in British Columbia, the number of fires varied six-fold and the area burned nearly a thousand-fold. Fewer fires now burn thirty-five times the area they did in the 1990s. A percolation threshold explains why, and a fire PDE on a patchy forest shows fuel 12.5% drier turning the same lightning strike into a fire a hundred times larger.
Read more →The Road That Made Everyone Late
Close a busy road and the traffic gets faster. Cities have done it and been surprised. The reason is arithmetic, it needs every driver to be perfectly selfish and perfectly informed, and in the smallest version a free new road costs everybody fifteen minutes. Then it turns out the most destructive road of all is not the free one.
Read more →The Urn That Decides Once
An urn with one red ball and one black. Draw one, put it back with a friend of the same colour, forever. The share of red settles — and it settles on a number drawn uniformly from nought to one, so there is no typical outcome at all. Then de Finetti removes the mechanism entirely, and one exponent decides whether the rich getting richer ends in a monopoly or in a permanent accident.
Read more →The Bird Does Not Come Home
A drunk man and a drunk pigeon leave the same pub with all the time in the world. The man is certain to find his way back. The pigeon manages it 34% of the time and otherwise never. One extra dimension turns a certainty into a bet you mostly lose — and then the sky turns out to have a ceiling.
Read more →One Hundred Prisoners, Fifty Opens
A hundred prisoners, a hundred boxes, fifty opens each, and all must succeed. Opening at random gives one chance in 1.27 x 10^30. One sentence of strategy gives 31 per cent, and it never decays with the size of the room.
Read more →A Coin With No Memory Still Looks Cold
For thirty years the hot hand was the textbook case of seeing pattern in noise. The test itself was biased: in four fair flips, the share of heads following a head averages 0.4048, not 0.5.
Read more →The Body That Is Always Falling
Stand still for a photograph and you are falling the whole time: an upside-down pendulum whose lean doubles every 0.22 seconds, caught by a brain that sees it late. The equation of standing is a second-order Langevin equation turned upside down, and it has a hard limit: with reflexes slower than 0.46 seconds, nothing can hold you up. Try it yourself.
Read more →Integrating Oscillations: Filon vs Levin
Two classical methods for oscillatory integrals that beat the O(omega)-points curse — compared against an exact solution, with the log-error plot to prove it.
Read more →The Stagflation Trap: Optimal Monetary Policy as an HJB Problem
A linear-quadratic HJB model that makes the stagflation trap mathematically precise: why a persistent supply shock forces the optimal steady state away from zero inflation and zero output gap simultaneously.
Read more →De Rerum Stochástica: Lucretius’s Random Swerve and the Birth of SDEs
Epicurus added the clinamen — a random, uncaused swerve — to save atomism from determinism. Two thousand years later, it became the diffusion term in every SDE ever written.
Read more →Markov Chains in Supermarket Management
How Markov chains model checkout queues and customer flow — and what the stationary distribution tells a supermarket manager about staffing and store layout.
Read more →The Kelly Criterion — Why the Optimal Strategy Is Never Used
A 60/40 coin. A provably optimal betting strategy. And almost nobody uses it. Kelly's 1956 result is one of the few places in finance where the correct answer is known and rationally ignored.
Read more →The Bookmaker’s Measure: Kelly, Martingale, and the Price of an Edge
The bookmaker is not running a gambling operation. They are constructing a risk-neutral measure. Understanding that changes everything about how you think about betting strategies — including why Martingale fails even when you have edge.
Read more →Four Historical Decisions an ABM Would Have Changed
Smoot-Hawley was not unique. From Versailles to Vietnam to Stalin's grain quotas to the Maginot Line — the same pattern recurs. A model that forgets the other agents in the system.
Read more →The Trade War Nobody Chose: Smoot-Hawley and the Model That Would Have Stopped It
In 1930, a thousand economists begged Hoover not to sign the Smoot-Hawley tariff. He signed it anyway. An agent-based model would have shown exactly what happened next — before it happened.
Read more →CAD 530 a Month: What Your Car Insurance Premium Actually Pays For
A friend bought a Ford F-150 in Ontario. The insurance quote came in at 530 dollars a month. I built a mathematical model to find out how much of that is justified.
Read more →Solving the 2D Wave Equation with Spectral Methods
Using the Dedalus spectral PDE solver to simulate the 2D wave equation on a periodic domain, with animated GIF output and applications across physics, finance, and medical imaging.
Read more →The Square Root That Keeps Rain Positive: CIR Processes in Weather Modelling
The Cox-Ingersoll-Ross process was built for interest rates. Its real strength — staying positive as noise vanishes near zero — turns out to be exactly what rainfall modelling needs.
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