ORBi Collection: Mathematics
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Pascal triangles and Sierpiński gasket extended to binomial coefficients of words
http://hdl.handle.net/2268/216574
Title: Pascal triangles and Sierpiński gasket extended to binomial coefficients of words
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<br/>Author, coauthor: Stipulanti, Manon
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<br/>Abstract: The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015 on a extension of the Pascal triangles to base2 expansions of integers. In a first part, I define two new objects that both generalize the classical Pascal triangle and the Sierpinski gasket. In a second part, I define a new sequence extracted from the Pascal triangle in base 2 and study its regularity. In a third part, I exhibit an exact formula for the behavior of the summatory function of the latter sequence.
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<br/>Commentary: Work in collaboration with Julien Leroy (ULg, j.leroy@ulg.ac.be) and Michel Rigo (ULg, m.rigo@ulg.ac.be). // Travail en collaboration avec Julien Leroy (ULg, j.leroy@ulg.ac.be) et Michel Rigo (ULg, m.rigo@ulg.ac.be).

From combinatorial games to shapesymmetric morphisms
http://hdl.handle.net/2268/216249
Title: From combinatorial games to shapesymmetric morphisms
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<br/>Author, coauthor: Rigo, Michel
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<br/>Abstract: The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words.
In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nimsum, SpragueGrundy function, Wythoff's game, ...). We also review some concepts from combinatorics on words. We thus introduce the wellknown kautomatic sequences and review some of their characterizations (in terms of morphisms, finiteness of their kkernel,...). These sequences take values in a finite set but the SpragueGrundy function of a game, such as Nim of Wythoff, is usually unbounded. This provides a motivation to introduce kregular sequences (in the sense of Allouche and Shallit) whose kkernel is not finite, but finitely generated.
In the second lecture, games played on several piles of token naturally give rise to a multidimensional setting. Thus, we reconsider kautomatic and kregular sequences in this extended framework. In particular, determining the structure of the bidimensional array encoding the (loosing) Ppositions of the Wythoff's game is a longstanding and challenging problem in CGT. Wythoff's game is linked to nonstandard numeration system: Ppositions can be determined by writing those in the Fibonacci system. Next, we present the concept of shapesymmetric morphism: instead of iterating a morphism where images of letters are (hyper)cubes of a fixed length k, one can generalize the procedure to images of various parallelepipedic shape. The shapesymmetry condition introduced twenty years ago by Maes permits to have welldefined fixed point.
In the last lecture, we move to generalized numeration systems: abstract numeration systems (built on a regular language) and link them to morphic (multidimensional) words. In particular, pictures obtained by shapesymmetric morphisms coincide with automatic sequences associated with an abstract numeration system. We conclude these lectures with some work in progress about games with a finite ruleset. This permits us to discuss a bit Presburger definable sets.

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport
http://hdl.handle.net/2268/216192
Title: General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport
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<br/>Author, coauthor: Machrafi, Hatim; Lebon, Georgy
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<br/>Abstract: A generalized heat transport equation applicable at small length and short time scales is proposed. It is based on extended irreversible thermodynamics with an infinite number of highorder heat fluxes selected as state variables. Extensions of Fick's and Ohm's laws are also formulated. As a numerical illustration, heat conduction in a rigid body subject to fixed and oscillatory temperature boundary conditions is discussed. © 2015 Elsevier Ltd.

Heat transfer at nanometric scales described by extended irreversible thermodynamics
http://hdl.handle.net/2268/216169
Title: Heat transfer at nanometric scales described by extended irreversible thermodynamics
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<br/>Author, coauthor: Machrafi, Hatim
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<br/>Abstract: The purpose of this work is to present a study on heat conduction in systems that are composed out of spherical and cylindrical microand nanoparticles dispersed in a bulk matrix. Special emphasis is put on the dependence of the effective heat conductivity on various selected parameters as particle size and also its shape, surface specularity and density, including particlematrix interaction. The heat transfer at nanometric scales is modelled using extended irreversible thermodynamics, whose main feature is to elevate the heat flux vector to the status of independent variable. The model is illustrated by a CopperSilicium (CuSi) system. It is shown that all the investigated parameters have a considerable influence, the particle size being especially useful to either increase or decrease the effective thermal conductivity. © 2016 Hatim Machrafi, licensee De Gruyter Open.

Special issue: Twelfth Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015)
http://hdl.handle.net/2268/215728
Title: Special issue: Twelfth Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015)
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<br/>Editor: Crama, Yves; Goossens, Dries; Leus, Roel; Schyns, Michael; Spieksma, Frits
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<br/>Abstract: This special issue of the Journal of Scheduling contains ten papers presented at the Twelfth Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015), held from June 8 to June 12, 2015, in La RocheenArdenne, Belgium.

Permutation groups and the MorseHedlund Theorem
http://hdl.handle.net/2268/215435
Title: Permutation groups and the MorseHedlund Theorem
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<br/>Author, coauthor: Charlier, Emilie
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<br/>Abstract: In this talk I presented the results of the paper "On a group theoretic generalization of the MorseHedlund theorem", which is a joint work with Svetlana Puzynina and Luca Zamboni. In this paper, we give a broad unified framework via group actions for constructing complexity functions of infinite words. Factor complexity, Abelian complexity and cyclic complexity are all particular cases of this general construction.

An open question about diametral dimensions
http://hdl.handle.net/2268/215199
Title: An open question about diametral dimensions
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<br/>Author, coauthor: Demeulenaere, Loïc
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<br/>Abstract: The diametral dimension is a topological invariant which characterizes Schwartz and
nuclear spaces. However, there exists another diametral dimension which was conjectured
by Bessaga, Mityagin, Pełczynski, and Rolewicz to be equal to the first one in
Fréchet spaces.
In this talk, we describe some conditions which assure the equality of the two diametral
dimensions in metrizable locally convex spaces. Besides, we explain why such an
equality is generally impossible in nonmetrizable spaces.

Théorèmes et problèmes de géométrie élémentaire
http://hdl.handle.net/2268/215128
Title: Théorèmes et problèmes de géométrie élémentaire
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<br/>Author, coauthor: Catalan, Eugène

Traité élémentaire de géométrie descriptive : atlas
http://hdl.handle.net/2268/215127
Title: Traité élémentaire de géométrie descriptive : atlas
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<br/>Author, coauthor: Catalan, Eugène

Ecologie du formalisme "bipoint" dans l'enseignement de la géométrie au niveau secondaire
http://hdl.handle.net/2268/215086
Title: Ecologie du formalisme "bipoint" dans l'enseignement de la géométrie au niveau secondaire
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<br/>Author, coauthor: Nguyen, Ngan Giang

Une approche heuristique d'une géométrie calculatoire
http://hdl.handle.net/2268/214926
Title: Une approche heuristique d'une géométrie calculatoire
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<br/>Author, coauthor: Nguyen, Ngan Giang; SchneiderGilot, Marguerite

An algorithm for computing nonconcave multifractal spectra using the Sν spaces
http://hdl.handle.net/2268/214879
Title: An algorithm for computing nonconcave multifractal spectra using the Sν spaces
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<br/>Author, coauthor: Kleyntssens, Thomas; Esser, Céline; Nicolay, Samuel
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<br/>Abstract: We present an implementation of a multifractal formalism based on the Sν spaces and show that it effectively gives the right Hölder spectrum in numerous cases. In particular, it allows to recover nonconcave spectra, where other multifractal formalisms only lead to the concave hull of the spectra.

Some notions of compactness in Functional Analysis and one related question about diametral dimensions
http://hdl.handle.net/2268/214791
Title: Some notions of compactness in Functional Analysis and one related question about diametral dimensions
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<br/>Author, coauthor: Demeulenaere, Loïc
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<br/>Abstract: The diametral dimension is a topological invariant on the class of topological vector spaces. Besides, there exists another diametral dimension which was claimed to be equal to the first one in Fréchet(Schwartz) spaces. However, this equality has never been proved. In this talk, we make some reminders about Functional Analysis and some related concepts (seminorms, locally convex spaces, etc.) and we insist on some notions linked to compactness. We also introduce the associated notions of Montel spaces and Schwartz spaces.
Then, we introduce the question about the equality of the two diametral dimensions and we explain why this is directly solved in nonMontel spaces and in Montel but not Schwartz spaces. Next, we describe some recent, positive, partial results in the context of Schwartz spaces. Finally, we explain what happens when we consider these diametral dimensions in nonmetrizable locally convex spaces.

Fully leafed induced subtrees
http://hdl.handle.net/2268/214654
Title: Fully leafed induced subtrees
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<br/>Author, coauthor: Blondin Massé, Alexandre; de Carufel, Julien; Goupil, Alain; Lapointe, Mélodie; Nadeau, Emile; Vandomme, Elise
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<br/>Abstract: Let G be a simple graph of n vertices. We consider the problem ISil of deciding whether there exists an induced subtree with exactly i ≤ n vertices and l leaves in G. We also study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by L_G(i), realized by an induced subtree with i vertices, for 2 ≤ i ≤ n. We compute the values of the map L_G for some classical families of graphs and in particular for the ddimensional hypercubic graphs Q_d , d ≤ 6. Then we prove that the ISil problem is in general NPcomplete. We also describe a nontrivial branch and bound algorithm that computes the function L_G for any simple graph G. In the special case where G is a tree, we provide a O(δn^3) time and O(n^2) space algorithm, where δ is the maximum degree of G. Finally, we exhibit a bijection between the set of discrete derivative of the sequences L_G (i) where G is caterpillar, and the set of prefix normal words.

Statistiques et méthodes de recherche (Biostatistiques)
http://hdl.handle.net/2268/214590
Title: Statistiques et méthodes de recherche (Biostatistiques)
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<br/>Author, coauthor: Vander Haegen, Marie

Les théorèmes d'incomplétude
http://hdl.handle.net/2268/214147
Title: Les théorèmes d'incomplétude
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<br/>Author, coauthor: Hoyoux, Renaud
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<br/>Abstract: Mémoire défendu en vue de l'obtention du titre de Master en sciences mathématiques portant sur les théorèmes d'incomplétude et constitué de quatre chapitres : le premier est un rappel des bases en logique formelle et en théorie des fonctions récursives; le second est consacré aux théorèmes d'incomplétude de Gödel; le troisième présente un exemple détaillé d'un énoncé aisé à comprendre mais néanmoins indémontrable dans l'arithmétique de Peano; le dernier explorant les possibilités de "mesure" des théorèmes indémontrables.

Regularized Discriminant Analysis in Presence of Cellwise Contamination
http://hdl.handle.net/2268/214039
Title: Regularized Discriminant Analysis in Presence of Cellwise Contamination
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<br/>Author, coauthor: Aerts, Stéphanie; Wilms, Ines
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<br/>Abstract: Quadratic and Linear Discriminant Analysis (QDA/LDA) are the most often applied classification rules under normality. In QDA, a separate covariance matrix is estimated for each group. If there are more variables than observations in the groups, the usual estimates are singular and cannot be used anymore. Assuming homoscedasticity, as in LDA, reduces the number of parameters to estimate. This rather strong assumption is however rarely verified in practice. Regularized discriminant techniques that are computable in highdimension and cover the path between the two extremes QDA and LDA have been proposed in the literature. However, these procedures rely on sample covariance matrices. As such, they become inappropriate in presence of cellwise outliers, a type of outliers that is very likely to occur in highdimensional datasets. We propose cellwise robust counterparts of these regularized discriminant techniques by inserting cellwise robust covariance matrices. Our methodology results in a family of discriminant methods that are robust against outlying cells, cover the gap between LDA and QDA and are computable in highdimension.

Impact of dependency on the distribution of pvalue
http://hdl.handle.net/2268/213661
Title: Impact of dependency on the distribution of pvalue
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<br/>Author, coauthor: Ernst, Marie; Swan, Yvik
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<br/>Abstract: We study the impact of dependence assumptions on the distribution of pvalues and quantiles for repeated testing on dependent data. This leads us to considering the general problem of the quality of a binomial approximation to the distribution of a sum of dependent indicator variables. Whenever possible we use classical and adhoc versions of Stein’s method to provide tight bounds on classical probability distances. In many cases, however, the relevant expressions are intractable and we resort to empirical analysis by extensive simulations. We apply our findings to a realistic reallife scenario.

Secondorder moment of the first passage time of a quasiHamiltonian oscillator with stochastic parametric and forcing excitations (under review)
http://hdl.handle.net/2268/213521
Title: Secondorder moment of the first passage time of a quasiHamiltonian oscillator with stochastic parametric and forcing excitations (under review)
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<br/>Author, coauthor: Vanvinckenroye, Hélène; Denoël, Vincent

Linearization and quadratization techniques for 01 optimization problems
http://hdl.handle.net/2268/213515
Title: Linearization and quadratization techniques for 01 optimization problems
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<br/>Author, coauthor: Rodriguez Heck, Elisabeth; Crama, Yves