Jellymatter is, we claim, not afraid of equations, but apparently scientists are. A study in PNAS claims to have found that theoretical biology papers are cited less when they are densely packed with mathematical language. The authors argue that this impedes progress, since empirical work needs to be backed up and commensurate with some theory to have deeper scientific meaning.
We’ve touched on the difference between chaos and randomness before. One strange property of chaotic systems is that they are able to synchronise to each other, so that in spite of their intrinsic tendency to vary wildly, a chaotic system can (actually quite easily) be persuaded to match the behaviour of another chaotic system. As this post will show, it is possible to use this property for a kind of secret message transmission.
The song, Twelve days of Christmas raises a number of important questions, like who gives milkmaids as presents? and will this song ever end? But most importantly, it makes us ask: If every day was a day of Christmas (like some may wish), would it still be physically possible to sing the song?
Here’s how long it took to sing each verse on the John Peel show one year when the great man was still alive:
I’ve seen some really bad methods for drawing confidence ellipsoids recently, they all seem to make it really complicated and confusing (and specific). So I thought I would show how to calculate points on an ellipse corresponding to a covariance matrix – this method works for any number of dimensions without any need to change it.
For all those that don’t care why, the method to generate the points of an ellipsoid is as follows:
1) make a unit n-sphere (which for 2D is a circle with radius 1), call these points X:
If it is an elipse you want, make a matrix with columns of and for some incrementing values between and (=开)
2) apply the following linear transformation to get the points of your ellipsoid (Y):
where M is the vector of the means (center of the ellipsoid) and is the covariance matrix. C represents the Cholesky Decomposition, sort of a matrix square root. k is the number of standard deviations at which one wishes to draw the ellipse
The Cholesky decomposition can be accessed as, “numpy.linalg.cholesky” in Python, “Cholesky” in R (matrix package), “chol” in MATLAB and “spotrf” (amongst others, I think) in LAPACK
For those who care, here is why this works…