Author Archives: PaulRS

Generating a random variable by coin tossing


1. Coin tossing Suppose we are given a sequence of independent fair bits (meaning take and with probabily ) we want to produce with them a discrete random variable that takes the values with probabilities . The objective, of course, … Continue reading

Posted in Algorithms, Number Theory, Probability | Leave a comment

Grading a class


Imagine we had a class with students and each week they hand in a sheet of homework. Now, you choose a set of of them at random and correct only those . At the end, the grade assigned to a … Continue reading

Posted in Asymptotics, Combinatorics, Computation, Probability | Leave a comment

Riemann-Lebesgue


The purpose of this post is to discuss the Riemann-Lebesgue Lemma, why it makes sense, and some interesting applications among which we have , the proof of which, in my opinion, explains a lot more clearly why it should equal … Continue reading

Posted in Computation | Leave a comment

Higher order Newton-Raphson


In this post we shall see one way to look at the Newton–Raphson formula, and the extensions this produces. Actually, I came up with this idea a few years back, while I was studying for a Numerical Analysis exam. Introduction. … Continue reading

Posted in Algorithms, Computation | Tagged , | Leave a comment

The fast decay of the gamma function


The purpose of this post is to discuss the fast decay of the gamma function when along the imaginary axis. This is an interesting topic, as it is not usually discussed, we are generally more interested in the increase of … Continue reading

Posted in Asymptotics, Computation | Leave a comment

The Fibonacci Word


Let us define by and , and extend it to a homomorphism of by . Let us note that Something very interesting is going on, each seems to be a prefix of . We will see that this is indeed … Continue reading

Posted in Combinatorics | Leave a comment

Partitions and divisors


The number of partitions of a non-negative integer is, roughly speaking, the number of ways in which we can write as a sum of positive integers disregarding the order of the terms, and we denote this number by . Formally … Continue reading

Posted in Combinatorics, Number Theory | Tagged , | Leave a comment