Solving an Unsolvable Math Problem

In “The Psychology of Invention in the Mathematical Field,” published in 1945, Jacques Hadamard quotes a mathematician who says, “It often seems to me, especially when I am alone, that I find myself in another world. Ideas of numbers seem to live. Suddenly, questions of any kind rise before my eyes with their answers.” In the back yard, Zhang had a similar experience. “I see numbers, equations, and something even—it’s hard to say what it is,” Zhang said. “Something very special. Maybe numbers, maybe equations—a mystery, maybe a vision. I knew that, even though there were many details to fill in, we should have a proof. Then I went back to the house.”

→ The New Yorker

Reality = Normal + Fat-Tail Distributions

To illustrate the phenomenon, consider the S&P’s daily percentage returns in terms of quantiles, which divides the performance record into equal-sized portions. The graph below plots the sample return of the S&P (black circles) against the theoretical quantiles (red line), defined here by a random distribution. If the S&P’s daily returns were perfectly random, the black circles would match the red line.


Normal distributions are still useful for analyzing markets and designing portfolios. Indeed, even in the daily return plot above it’s clear that the distribution looks quite normal for a fair amount of the sample. We can’t rely on normality alone for modeling markets. Factoring in fat-tails risk is essential. But letting a fat-tail worldview dominate your analysis is every bit as flawed as assuming that normal distributions will prevail. Asset pricing doesn’t neatly fit into one theoretical box, which means that our analytical tool kit shouldn’t be in a conceptual straightjacket either.

→ The Capital Spectator

To Predict Turbulence, Just Count the Puffs

It’s not that the stakes are low. A thorough explication of turbulence in pipes could help illuminate the transition to turbulence in a wide range of settings. Understanding how to minimize turbulence in air and fluids could ultimately help engineers pump oil through long pipelines more efficiently and build cars that generate less wind resistance. It could also allow them to harness turbulence more effectively in the settings in which it is helpful, as when vortices near an airplane wing pull a smooth layer of air toward the wing and allow the plane to come in for a slower and gentler landing.

→ Nautilus