2+ day, 1+ hour ago  (248+ words) The previous post linked to a post I wrote a few years ago about the Hilbert transform and Fourier series. That post says that if the Fourier series of a function is then the Fourier series of its Hilbert transform…...

2+ day, 2+ hour ago  (231+ words) The previous post announced some notes I wrote up based on an article by Henry Baker implementing functions of a complex variable in terms of functions of a real variable. That is, it finds functions u(x, y) and v(x,…...

2+ day, 13+ hour ago  (233+ words) A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations You can do something analogous for all the elementary functions, though some…...

3+ day, 22+ hour ago  (406+ words) You can't always get what you want. But sometimes you get what you need." " The Rolling Stones Circular functions and hyperbolic functions aren't invertible, but we invert them anyway. These functions map many points in the domain to each point…...

4+ day, 36+ min ago  (196+ words) The difference between a circular function and a hyperbolic function is a rotation or two. For example, cosh(z) = cos(iz). You can read that as saying that to find the hyperbolic cosine of z, first you rotate'z a quarter turn…...

5+ day, 23+ hour ago  (342+ words) Markov numbers are integer solutions to The Wikipedia article on Markov numbers mentions that Don Zagier studied Markov numbers by looking the approximating equation x" +y" +z" = 3xyz+ 4/9 which is equivalent to wheref(t) is defined as arccosh(3t/2). It wasn't clear to…...

1+ week, 3+ day ago  (254+ words) I've written a couple posts lately about reverse engineering the internal state of a random number generator, first Mersenne Twister then lehmer64. This post will look at xorshift128, implemented below. Recovering the internal state of the generator is simple: it's the four…...

1+ week, 6+ day ago  (207+ words) If you look very closely at my previous post, you'll notice that I initialize a 128-bit integer with a 64-bit value. The 128-bit unsigned integer represents the internal state of a random number generator. Why not initialize it to a…...

1+ week, 6+ day ago  (330+ words) A couple days ago I wrote about hacking the Mersenne Twister. I explained how to recover the random number generator's internal state from a stream of 640 outputs. This post will do something similar with the lehmer64 random number generator. This generator…...

1+ week, 6+ day ago  (282+ words) This morning I wrote a post about the probability that a random matrix over a finite field is invertible. If the field has q elements and the matrix has dimensions n " n then the probability is In that post I…...