arXiv subjects

Guénolé C. M. Silvestre

Publications and source records attributed to Guénolé C. M. Silvestre.

4 recordsLinked to original sources

arXiv2026

Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings

We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same enumerations when runs of ones and runs of zeros are jointly considered. We give the connections between these two types of run enumeration, and with the problem of compositions. We also analyse the same enumerations with a Hamming weight constraint. We discuss which of the many number sequences that emerge from these problems are already known and listed in the OEIS. Additionally, we extend our main enumerative results to the probabilistic scenario in which binary strings are outcomes of independent and identically distributed Bernoulli variables.

math.CO↗

arXiv2023

Runs of Ones in Binary Strings

We give three different computations of the total number of runs of length $i$ in binary $n$-strings, and we discuss the connection of this problem with the compositions of $n$.

math.CO↗

arXiv2023

General Distributions of Number Representation Elements

We provide general expressions for the joint distributions of the $k$ most significant $b$-ary digits and of the $k$ leading continued fraction coefficients of outcomes of an arbitrary continuous random variable. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the $j$-th significant digit, which is the counterpart of the general convergence law of the distribution of the $j$-th continued fraction coefficient (Gauss-Kuz'min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among other results. The particularisation for Pareto variables -- which include Benford variables as a special case -- is specially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading continued fraction coefficients of real data. Our results may find practical application in all areas where Benford's law has been previously used.

math.PR↗

arXiv2022

A Transformer Architecture for Online Gesture Recognition of Mathematical Expressions

The Transformer architecture is shown to provide a powerful framework as an end-to-end model for building expression trees from online handwritten gestures corresponding to glyph strokes. In particular, the attention mechanism was successfully used to encode, learn and enforce the underlying syntax of expressions creating latent representations that are correctly decoded to the exact mathematical expression tree, providing robustness to ablated inputs and unseen glyphs. For the first time, the encoder is fed with spatio-temporal data tokens potentially forming an infinitely large vocabulary, which finds applications beyond that of online gesture recognition. A new supervised dataset of online handwriting gestures is provided for training models on generic handwriting recognition tasks and a new metric is proposed for the evaluation of the syntactic correctness of the output expression trees. A small Transformer model suitable for edge inference was successfully trained to an average normalised Levenshtein accuracy of 94%, resulting in valid postfix RPN tree representation for 94% of predictions.

cs.CL↗