A while back I developed a mild obsession with pentagons (mathematical ones, not symbolistic!) It started when I discovered some beautiful (simple and to me, unknown) theorems of quadrangles, such as
Pentagons
Update to Functional Equations Reference (version 1.3)
This is a substantial update of this reference document. The most important addition is the chain and substitution rules for arithmetic difference calculus (ADC). Other additions include: more properties of the discrete power function, more properties of ADC operators, definitions of…
Update: Reference for Functional Equations
In this new version of Reference for Functional Equations I added several more z-transform pairs. I also started to add binomial transform pairs. The definition for the binomial is not consistent among different authors. I arbitrarily chose one, and later I changed…
Generating Random Integers With Arbitrary Probabilities
I finally laid my hands on Donald Knuth’s The Art of Computer Programming (what a wonderful set of books!), and found a neat algorithm for generating random integers 0, 1, 2, … , n – 1, with probabilities p_0, p_1,…
Estimating a Continuous Distribution from a Sample Set
It is sometimes necessary to find the distribution given a sample set from that distribution. If we do not know anything about the distribution, we cannot recover it exactly, so here we look at ways of finding a (discrete) approximation.…
Generating Random Points from Arbitrary Distributions for 2D and Up
I have already covered how to generate random numbers from arbitrary distributions in the one-dimensional case. Here we look at a generalisation of that method that works for higher dimensions. The basic trick, while easy to understand, is hard to…
A Reference for Functional Equations
I have not posted in a while; one reason is that I got sucked into some interesting mathematics; the work-in-progress Reference for Functional Equations is the result. If you are interested in such things – have a look. Bookmark on Delicious…
Generating Random Numbers with Arbitrary Distributions
For many applications, detailed statistical models are overkill. Instead, we can get away with a rough description of the distribution – not in mathematical formula form, but just as a graph with a few sample points. For example, when trying…
