Greek ligature -ger- (γερ) in medieval minuscule handwriting Fourier transform. Recall the formula \[ \int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}tx}\tfrac{1}{\sqrt{2\pi\sigma^2}}\mathrm{e}^{-\frac{x^2}{2\sigma^2}}\mathrm{d}x =\mathrm{e}^{-\frac{\sigma^2 t^2}{2}}. \] A good way to remember that the dispersion parameter $\sigma^2$ is in the numerator in the right hand side is to have in mind the uncertainty principle : if the signal is localized then its Fourier transform is not, and vice versa. In case of doubt regarding the sign inside the right hand side, remember that the standard Gaussian is an ..read more

Nicolas Léonard Sadi Carnot (1796 – 1932) A romantic figure behind entropy Why entropy emerges in basic mathematics? This tiny post aims to provide some answers. Combinatorics. Asymptotic analysis of the multinomial coefficient via the Stirling formula (itself a consequence of the Laplace method on the Euler Gamma function) : \[ \frac{1}{n}\log\binom{n}{n_1,\ldots,n_r} \xrightarrow[n=n_1+\cdots+n_r\to\infty]{\nu_i=\frac{n_i}{n}\to p_i} \mathrm{S}(p):=-\sum_{i=1}^rp_i\log(p_i). \] The multinomial coefficient is interpreted as the number of microstates compatible with a macrostate $(n_1,\ldots,n ..read more

The Fields Medal and its portrait of Archimedes. Archimedes theorem. Archimedes of Syracuse (-287 – -212) is one of the greatest minds of all times. One of his discoveries is as follows : if we place a sphere in the tightest cylinder, then the surface of the sphere and of the cylinder are the same, and moreover this remains valid if we cut the whole by a perpendicular plane. Archimedes was so proud of this theorem that he put the picture of it on his tombstone. This picture on the stone allowed his admirer Marcus Tullius Cicero (-106 – -43) to identify the tomb, in the year -75, almost one cen ..read more

Cheikha Remitti (1923 – 2006) Cheb Khaled (1960 – ) est peut-être le chanteur de raï le plus connu. Son album le plus réussi est sans doute Kutché (1988), en collaboration avec Safy Boutella (1950 – ). La plupart des chanteurs de raï de la génération de Khaled, et en particulier Khaled lui-même, notamment dans Kutché, ont été influencés par Cheikha Remitti (1923 – 2006). Connaissez-vous cette grand-mère et reine du raï ? On trouve chez elle la même puissance existentielle que dans le Delta blues du Mississippi. En voici une version modernisée tardive : La plus aristocratique Taos Amrouche (19 ..read more

Cheikha Remitti (1923 – 2006) Cheb Khaled (1960 – ) est peut-être le chanteur de raï le plus connu à l’international. L’album que je préfère de lui est Kutché (1988), en collaboration avec Safy Boutella (1950 – ). C’est l’un des tous meilleurs albums du siècle. La plupart des chanteurs de raï de la génération de Khaled, et en particulier Khaled lui-même, notamment dans Kutché, ont été influencés par Cheikha Remitti (1923 – 2006). Connaissez-vous cette grand-mère et reine du raï ? On trouve chez elle la même puissance existentielle que dans le blues des USA. En voici une version tardive moderne ..read more

1820 watercolor caricatures of Adrien-Marie Legendre (left) and Joseph Fourier by Julien-Léopold Boilly. This post is about a link between large deviations inequalities and the central limit phenomenon. In a nutshell, for the random walk $S_n:=X_1+\cdots+X_n$ where $X_1,\ldots,X_n$ are independent copies on a real random variables $X$, the Markov inequality yields, for all $x$, $$ \mathbb{P}\Bigr(\frac{S_n}{n}\geq x\Bigr) \overset{\lambda>0}{=} \mathbb{P}(\mathrm{e}^{\lambda S_n}\geq\mathrm{e}^{n\lambda x}) \leq\mathrm{e}^{-n\sup_{\lambda\geq0}(\lambda x-\log\mathbb{E}(\mathrm{e}^{\lambda X ..read more

1820 watercolor caricatures of Adrien-Marie Legendre (left) and Joseph Fourier by Julien-Léopold Boilly. This post is about a link between large deviations inequalities and the central limit phenomenon. In a nutshell, for the random walk $S_n:=X_1+\cdots+X_n$ where $X_1,\ldots,X_n$ are independent copies on a real random variables $X$, the Markov inequality yields, for all $x$, $$ \mathbb{P}\Bigr(\frac{S_n}{n}\geq x\Bigr) \overset{\lambda>0}{=} \mathbb{P}(\mathrm{e}^{\lambda S_n}\geq\mathrm{e}^{n\lambda x}) \leq\mathrm{e}^{-n\sup_{\lambda>0}(\lambda x-\log\mathbb{E}(\mathrm{e}^{\lambda X ..read more

Laurent Saloff-Coste (1958 – ) Laurent Saloff-Coste was one of my professors during my second year of Master. His teaching style was truly original, not at all academic, difficult to reproduce. I greatly appreciated his course on random walks on groups, between probability and algebra, involving analysis thru Sobolev inequalities and geometry thru isoperimetry. This post is about the cutoff phenomenon for ergodic Markov processes, more precisely the analysis conducted by Laurent Saloff-Coste and his student Guan-Yu Chen in 2008 for the \( {\max-L^p} \) cutoff, \( {1<p<\infty} \) under th ..read more

Freeman Dyson (1923 – 1920) – Explorer of links between random matrix ensembles and log-gases The McCarthy multimatrix ensemble of random matrices. For all integers $n\geq1$ and $d\geq1$, let $\mathbb{M}_{n,d}$ be the set of $d$-tuples $(M_1,\ldots,M_d)$ of $n\times n$ Hermitian matrices such that \[ M_jM_k=M_kM_j\quad\text{for all $1\leq j,k\leq d$.} \] We equip this hypersurface with the trace of the Gaussian distribution, namely \[ (M_1,\ldots,M_d) \mapsto\frac{1}{Z_{n,d}} \mathrm{e}^{-\sum_{k=1}^d\mathrm{Tr}(M_k^2)}\mathrm{d}M \] where $\mathrm{d}M$ is the trace on $\mathbb{M}_{n,d}$ of th ..read more

Caricature of Camille Jordan (1838 – 1922) by one of his students at École Polytechnique, during a lecture on differential calculus : « J’ai mis ∂u mais c’est une faute d’impression ». This tiny post is about stability in the sense of Lyapunov of stationary points of autonomous ordinary differential equation (ODE) in $\mathbb{R}^n$ given by \[ x'(t)=f(x(t)),\quad x(0)=x_0\] where $f:O\subset\mathbb{R}^n\to\mathbb{R}^n$ is a $\mathcal{C}^1$ vector field. This includes two important examples: Gradient flows : $f=-\nabla\mathcal{E}$ where $\mathcal{E}:O\to\mathbb{R}$ is $\mathcal{C}^2$ Geometric ..read more