# How Far Apart Are Two Random Points in a Hypercube?3

Posted by Cleve Moler,

Two days ago I wrote about random points in a square. At the last minute I added the paragraph asking about the generalization to random points in a cube. I have to admit that I didn't check the Web to see what was known about the question.... read more >>

# How Far Apart Are Two Random Points in a Square?5

Posted by Cleve Moler,

How far apart can you expect two points chosen at random in the unit square to be? I found this problem on the YouTube channel maintained by Presh Talwalkar, Mind Your Decisions. He correctly calls it a very hard puzzle. At first, I guessed the answer might be $1/2$. But the correct answer is more interesting than that.... read more >>

# Fitting and Extrapolating US Census Data

Posted by Cleve Moler,

A headline in the New York Times at the end of 2016 said "Growth of U.S. Population Is at Slowest Pace Since 1937". This prompted me to revisit an old chestnut about fitting and extrapolating census data. In the process I have added a couple of nonlinear fits, namely the logistic curve and the double exponential Gompertz model.... read more >>

# My Favorite ODE

Posted by Cleve Moler,

My favorite ordinary differential equation provides a good test of ODE solvers, both numeric and symbolic. It also provides a nice illustration of the underlying existence theory and error analysis.... read more >>

# Modernization of Numerical Integration, From Quad to Integral

Posted by Cleve Moler,

The MATLAB functions for the numerical evaluation of integrals has evolved from quad, through quadl and quadgk, to today's integral. ... read more >>

# Strang and Moler Video Course on Differential Equations

Posted by Cleve Moler,

Gil Strang has produced a MOOC-style video course on Differential Equations and Linear Algebra. I have added some videos about the MATLAB ODE suite. The series is available from the MathWorks Web site, MIT OpenCourseWare and several other popular sources.... read more >>

# Tumbling Box ODE2

Posted by Cleve Moler,

A rectangular box, such as a book or a cell phone, thrown in the air can tumble stably about its longest axis, or about its shortest axis, but not about its middle axis.... read more >>

# Discover e with a graphical experiment

Posted by Cleve Moler,

An interactive graphical experiment lets you discover the value of one of the most important numerical quantities in mathematics. ... read more >>

# Periodic Solutions to the Lorenz Equations

Posted by Cleve Moler,

Changing the value of a parameter in the equations that produce the famous Lorenz chaotic attractor yields nonlinear ordinary differential equations that have periodic solutions. ... read more >>

# Complex Step Differentiation3

Posted by Cleve Moler,

Complex step differentiation is a technique that employs complex arithmetic to obtain the numerical value of the first derivative of a real valued analytic function of a real variable, avoiding the loss of precision inherent in traditional finite differences.... read more >>