Math blog of Samuel Alexandre Vidal

Café Math

Hi, welcome to my small Math blog. I hope you'll enjoy the content. I'll try to post on a regular basis some cool math stuff.

As you probably know, the Universe of Mathematics is filled with superb landscapes, clear and deep waters, abundant life forms, never ending ramifications and amazing correspondences. The intent of this blog is to serve as a cosy café to take a break and discuss some cool mathematical topics.

Weighting Bags

Thu, 11 Sep 2013 01:09:11 GMT

Hi, today my colleague Stuart told me about the following brain teaser of combinatorics. I've found it pretty interesting. Problem. Suppose you have twelve bags and a scale. All of the bags weight the same, except for one which is either lighter or heavier than the others. You have to determine which is that bag and if it is heavier or lighter in only three weightings. [Read more...]

Dual Numbers and Markov Chains (Part 1)

Thu, 09 Sep 2013 19:50:16 GMT

Hi, today Alfredo and I want to talk about dual numbers and how to apply them to markov chain problems. A dual number is an expression of the form, $$a + b\varepsilon,$$ where $a$ and $b$ are elements of a given ring (real, complex, matrices or anything) and where $\varepsilon^2 = 0$. The numbers $a$ and $b$ are its real part and dual part, respectively. This is analogous to complex numbers $a + i b$ where $i^2 = -1$ but contrary to the complex numbers, dual numbers do not constitute a field. The rules of computation are as follows. [Read more...]

Basic Power Series Algorithms

Thu, 05 Sep 2013 22:10:07 GMT

Hi, the other day I wanted to do some formal series computations for my next research article and I was tiered of using existing computer algebra programs. I decided to implement some fast formal power series algorithms in C#, just for fun. Today, I want to share what I came up with. [Read more...]

Grassmannians

Sat, 31 Aug 2013 15:34:32 GMT

Hi, today is my birthday, I'm 32 years old. It's been more than a year since my last post on this blog. For this new post I would like to talk about grassmannian varieties and Schubert cell decomposition of those varieties. Along the way we are going to talk about the quantum binomial coefficients, the Young diagrams and some related pretty combinatorics. [Read more...]

Arithmetic Functions (Part 1)

Thu, 31 May 2012 23:56:12 GMT

In the last blog article we touched a word on the Möbius function $\mu$ and the Möbius inversion formula. Another interesting way to see the topic is through arithmetic multiplicative functions. Today I want to give an account of some of the most basic notions and results of this last subject. I hope you'll enjoy reading it as much as enjoyed writing it. [Read more...]

A Tale of Multiple Zeta Values (Part 1)

Mon, 21 May 2012 19:46:32 GMT

Hi, today I want start a series of posts on some of the most puzzling questions of number theory. The guiding theme will be that of \emph{multiple zeta values}. They are particular numbers which are deeply engraved in lots of the research of the last thirty years. And they are a recurring them in the deep connections that are discovered between combinatorics, mathematical physics, higher arithmetics (for example the structure of the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$, low dimension topology (like the classification of knots, links etc...). Despite of their importance, very little is known about those numbers arithmetically, and most of the theory is conjectural. Due to their ubiquity, any significant progress in the understanding of those numbers is guaranteed to have far reaching consequences. [Read more...]

A Question of (First) Principles

Sun, 13 May 2012 6:16:51 GMT

Yesterday my friend Fiza told me about an annoying question that her students asked her, namely, Can we prove that minus multiplied by minus gives plus ? Mmm... that's the kind of question with no clear answer. As this is merely a postulate... But the conversation took an interesting turn as we talked about how to construct the negative numbers from the non-negative ones. [Read more...]

Euler-MacLaurin Formula

Tue, 8 May 2012 4:40:17 GMT

Hi, today I want to talk about one of my favorite formula in Mathematics, called the Euler-MacLaurin Formula. $$ \frac{1}{2}\left[\, Q(n) + Q(0) \,\right] + \sum_{k = 1}^{n-1} Q(k) = \int_0^n Q(x)\,dx + R_n, $$ with, $$ R_n = \sum_{k = 1}^\infty \frac{B_{2k}}{(2k)!} \,\left[\, Q^{(2k-1)}(n) - Q^{(2k-1)}(0) \,\right]. $$ It can be interpreted as a series expression of the reminder term $R_n$ of the $n$-step integration using trapezoid method. [Read more...]

Geometric Series

Mon, 23 Apr 2012 11:42:27 GMT

Hi in this post is for my friend Mrigank who wants to learn about geometric series. His question was: How to understand the following picture ? [Read more...]

Computing $\alpha$-Stable Densities Using the FFT

Mon, 9 Apr 2012 11:45:28 GMT

Yesterday I reopened a box of handwritten documents that I wrote when I had the time to study and think about math $24/7$. Those boxes were packed when I moved from Lille to Paris almost two years ago and remained unopened since then sitting there near my shelf collecting the dust. I am really happy to have my hands on those gems again and today, I'll talk about a small and clever computation that I've found inside that box while scanning quickly through them, namely: how to compute the density function of an $\alpha$-stable distribution using the fast Fourier transform algorithm. [Read more...]

Monte Carlo Integration

Sun, 8 Apr 2012 6:19:15 GMT

Hi today, I want to talk about the basic aspects of Monte Carlo integration. It is a stochastic algorithm to numerically compute an integral, $$ \int_I f(x) \, dx $$ It has as host of good properties. First it is surprisingly simple, second, the exact same method can be applied whatever the function $f$ is, in whatever dimension. It doesn't get more complicated as $f$ gets complicated or as the dimensionality of the problem changes. Its main disadvantage over other methods is that it is spectacularly slow. [Read more...]

Black-Scholes Equation

Sun, 1 Apr 2012 22:09:27 GMT

Hi today I want to talk about the most famous equation of quantitative finance, the celebrated and Nobel prize winner known as the Black-Scholes-Merton equation, $$ \frac{\partial}{\partial t} V + rS \,\frac{\partial}{\partial S} V + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2}{\partial S^2} V = r V $$ [Read more...]

Adjacency Matrices of Graphs

Wen, 29 Mar 2012 01:06:32 GMT

Yesterday I rewatched the movie "Good Will Hunting" with Matt Damon. The film is really good, if you haven't saw it yet, go see it. It really is a masterpiece. It tells the story of some genius orphan young man that was detected by a mathematician as he solved problems that were written on a black board at the University where he was employed as a cleaner. By chance there is one second in the film when one can see the problem written on the black board. It is an actual math problem. Not that difficult but by no mean trivial. [Read more...]

Moving Averages

Wen, 14 Mar 2012 00:06:52 GMT

Hi today I want to talk about moving averages and their practical computation. The questions treated here were asked to me as part of quant job interview one year ago. Here is a transcription of my responses. If you are curious... I got the job. [Read more...]

Optimal Execution of Portfolio Transactions

Sat, 3 Mar 2012 15:07:22 GMT

Hi, today I want to talk about the article \emph{Optimal Execution of Portfolio Transactions} by R. Almgren and N. Chriss \emph{Journal of Risk}, 3(2):5-39, 2000 which I found genuinely interesting. The paper is full of insightful remarks. It provides a simple model used to describe optimal execution of portfolio transactions. The idea is to segment the execution of a big transaction into smaller trades at successive time intervals in order to reduce the execution costs in the context of an insufficiently liquid market. This is one of the primary goals of algorithmic trading. [Read more...]

Basic Definitions in Category Theory

Sun, 26 Feb 2012 19:15:15 GMT

Hi, some times ago, I wrote a concise description of the very basic concepts of category theory. Unfortunately the text was lost until very recently. I'm very happy have it again and I post it here so that some other people can see it. [Read more...]

Itô Calculus (Part 1)

Sun, 19 Feb 2012 19:22:15 GMT

Today I want to talk about something very useful called Itô calculus. The stochastic differential calculus invented by Itô is a way to extend the ordinary differential calculus by extending the rules a bit, so that it can handle computations involving stochastic processes. [Read more...]

Brownian Motion (Part 1)

Mon, 6 Feb 2012 00:44:35 GMT

Hi, today I want to talk about the important concept of brownian motion. In the first part I will recall what is a brownian motion and what are their basic properties. After that we'll look at practical ways to simulate them. [Read more...]

Cassipoeia Type System (Part 1)

Sat, 28 Jan 2012 20:45:01 GMT

Cassiopeia is a small computer algebra system I am writing. It is still very basic, but it is aimed at having a rich a powerful type system. This article is the first of a series on the theory behind that type system. The goam is to provide a conveinent dictionary between some sophisticated computer software type system (like that of C#) and the theory of categories which I chose as a foundation of the mathematical engine. [Read more...]

An Incomplete Elliptic integral

Fri, 27 Jan 2012 00:30:01 GMT

Yesterday my friend Fiza asked me about an interesting mathematical problem. She stumbled across this problem while doing her research on cosmology. she wanted to compute the integral, $$ \int_0^x \frac{dx}{\sqrt{\cos x}} $$ As elementary as it may look, it turned out to be very cool mathematically. [Read more...]

Cauchy-Riemann equation

Sat, 22 Jan 2012 06:34:01 GMT

Today I want to expand on a remark one of us made about holomorphic functions. What are holomorphic functions, why are they so stunning, what makes them so useful ? [Read more...]

Quantitative Finance

Some thoughts on modeling financial markets with math. (Comming soon)

Black-Scholes formula

Sat, 22 Jan 2012 03:06:01 GMT

(Comming soon)