ORBi Collection: Mathematics
http://hdl.handle.net/2268/153
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The sequential analogue of the hypergeometric law
http://hdl.handle.net/2268/197326
Title: The sequential analogue of the hypergeometric law
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<br/>Author, co-author: Esch, LouisUn théorème de la théorie de la mesure
http://hdl.handle.net/2268/197321
Title: Un théorème de la théorie de la mesure
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<br/>Author, co-author: Esch, LouisModèles stochastiques de la thrombocytopoïèse
http://hdl.handle.net/2268/197263
Title: Modèles stochastiques de la thrombocytopoïèse
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<br/>Author, co-author: Esch, LouisLe problème du sous-espace invariant
http://hdl.handle.net/2268/197108
Title: Le problème du sous-espace invariant
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<br/>Author, co-author: Demeulenaere, Loïc
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<br/>Abstract: La présentation débute par la définition du problème du sous-espace invariant. Après quelques cas simples, elle aborde l'historique de ce problème et donne plusieurs résultats déjà connus.
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<br/>Commentary: Il s'agit d'un compte-rendu présenté lors d'un séminaire d'Analyse et concernant un sujet abordé par Etienne Matheron lors du congrès organisé en l'honneur des 60 ans de Gilles Godefroid (Mons, 4 et 5 novembre 2013).Diametral Dimension of topological vector spaces
http://hdl.handle.net/2268/197097
Title: Diametral Dimension of topological vector spaces
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<br/>Author, co-author: Demeulenaere, Loïc
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<br/>Abstract: We present the definition and the main properties of the diametral dimension. We give an application to prove that two spaces are non-isomorphic. We also consider the case of Snu spaces.Implementing and Comparing Stochastic and Robust Programming
http://hdl.handle.net/2268/197090
Title: Implementing and Comparing Stochastic and Robust Programming
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<br/>Author, co-author: Cuvelier, Thibaut
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<br/>Abstract: Traditional optimisation tools focus on deterministic problems: scheduling airline flight crews (with as few employees as possible while still meeting legal constraints, such as maximum working time), finding the shortest path in a graph (used by navigation systems to give directions, usually based on GPS signals), etc.
However, this deterministic hypothesis sometimes yields useless solutions: actual parameters cannot always be known to full precision, one reason being their randomness. For example, when scheduling trucks for freight transportation, if there is unexpected congestion on the roads, the deadlines might not be met, the company might be required to financially compensate for this delay, but also for the following deliveries that could not be made on schedule.
Two main approaches are developed in the literature to take into account this uncertainty: take decision based on probability distributions of the uncertain parameters (stochastic programming) or considering they lie in some set (robust programming). In general, the first one leads to a large increase in the size of the problems to solve (and thus requires algorithms to work around this dimensionality curse), while the second is more conservative but tends to change the nature of the programs (which can impose a new solver technology).
Some authors [2] claim that those two mindsets are equivalent, meaning that the solutions they provide are equivalent when faced with the same uncertainty. The goal of this thesis is to explore this question: for various problems, implement those two approaches, and compare them.
Is one solution more secluded from variations due to the uncertain parameters?
Does it bring benefits over a deterministic approach?
Is one cheaper than the other to compute?Optimisation and uncertainty: comparing stochastic and robust programming
http://hdl.handle.net/2268/197081
Title: Optimisation and uncertainty: comparing stochastic and robust programming
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<br/>Author, co-author: Cuvelier, Thibaut
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<br/>Abstract: Traditional optimisation tools focus on deterministic problems: scheduling airline flight crews (with as few employees as possible while still meeting legal constraints, such as maximum working time), finding the shortest path in a graph (used by navigation systems to give directions), etc.
However, this deterministic hypothesis sometimes provides useless solutions: actual parameters cannot always be known to full precision, one reason being their randomness. For example, when scheduling trucks for freight transportation, if there is unexpected congestion on the roads, the deadlines might not be met, the company might be required to financially compensate for this delay, but also for the following deliveries that could not be made on schedule.
Two main approaches are developed in the literature to take into account this uncertainty: make decision based on probability distributions of the uncertain parameters (stochastic programming) or considering they lie in a so-called ¿uncertainty set¿ (robust programming). In general, the first one leads to a large increase in the size of the problems to solve (and thus requires algorithms to work around this dimensionality curse), while the second is more conservative but tends to change the nature of the programs (which can impose a new solver technology). This talk compares the two approaches on three different cases: facility location, unit-commitment, and reservoir management.
On the implementation side, multiple specific algorithms have been implemented to solve stochastic programs in order to compare their relative performance: Benders¿ decomposition, progressive hedging, and the deterministic equivalent. When comparing stochastic and robust programming, many differences appear in many aspects, even though the literature about those is very scarce. (Furthermore, those two approaches are not incompatible: both can be used in the same optimisation model to take into account different parts of the uncertainty.)
Concerning solving time, stochastic programming quickly gives rise to intractable problems, which requires the development of more specific algorithm just to be able to solve them to an acceptable accuracy in decent time. What is more, the stochastic description of the uncertain values (with a discretisation of the probability distribution through scenarios) must cover all the possible uncertainty, otherwise the solution risks overfitting those scenarios, and is likely to have poor performance on close but different scenarios that may happen in practice ¿ which imposes to use a large number of scenarios, which yields very large (and hardly tractable) optimisation programs.
On the other hand, by using specific uncertainty sets, robust programming yields programs that are only very slightly harder to solve, with an objective function that is very close to that of stochastic programming, but with totally different robustness properties: by using an uncertainty set computed from the scenarios, and not the scenarios themselves, it is able to withstand a much higher uncertainty than stochastic programming. However, when facing other types of uncertainty, this conclusion might turn untrue, with robust programming unable to cope with them and to bring interesting solutions to the table.Extrait d'une lettre adressée à M. Hermite
http://hdl.handle.net/2268/197017
Title: Extrait d'une lettre adressée à M. Hermite
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<br/>Author, co-author: Catalan, EugèneDémonstration d'une propriété sur les courbes (solution de la question 311)
http://hdl.handle.net/2268/197009
Title: Démonstration d'une propriété sur les courbes (solution de la question 311)
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<br/>Author, co-author: Catalan, EugèneRemarque sur une note de M. Latars
http://hdl.handle.net/2268/197000
Title: Remarque sur une note de M. Latars
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<br/>Author, co-author: Catalan, EugèneUne généralisation du triangle de Pascal
http://hdl.handle.net/2268/196910
Title: Une généralisation du triangle de Pascal
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<br/>Author, co-author: Stipulanti, Manon; Rigo, Michel; Leroy, Julien
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<br/>Abstract: We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1]×[0, 1] associated with this extended Pascal triangle modulo a prime p.Une généralisation du triangle de Pascal et la suite A007306
http://hdl.handle.net/2268/196909
Title: Une généralisation du triangle de Pascal et la suite A007306
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<br/>Author, co-author: Stipulanti, Manon
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<br/>Abstract: We introduce a generalization of Pascal triangle based on bino- mial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1]×[0, 1] associated with this extended Pascal triangle modulo a prime p. We consider a sequence (S(n))n≥0 counting the number of positive entries on each row of the generalized Pascal triangle. By introducing a convenient tree structure, we provide a recurrence relation for (S(n))n≥0, we prove that the sequence is 2-regular, give a linear representation and make the connection with the sequence of denominators occurring in the Farey tree.Generalized Pascal triangle for binomial coefficients of finite words
http://hdl.handle.net/2268/196908
Title: Generalized Pascal triangle for binomial coefficients of finite words
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<br/>Author, co-author: Stipulanti, Manon; Leroy, Julien; Rigo, Michel
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<br/>Abstract: Abstract. We introduce a generalization of Pascal triangle based on bino- mial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p.A class of valid inequalities for multilinear 0-1 optimization problems
http://hdl.handle.net/2268/196789
Title: A class of valid inequalities for multilinear 0-1 optimization problems
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<br/>Author, co-author: Crama, Yves; Rodriguez Heck, Elisabeth
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<br/>Abstract: This paper investigates the polytope associated with the classical standard
linearization technique for the unconstrained optimization of multilinear
polynomials in 0-1 variables. A new class of valid inequalities, called
2-links, is introduced to strengthen the LP relaxation of the standard linearization.
The addition of the 2-links to the standard linearization inequalities
provides a complete description of the convex hull of integer solutions
for the case of functions consisting of at most two nonlinear monomials.
For the general case, various computational experiments show that the 2-
links improve both the standard linearization bound and the computational
performance of exact branch & cut methods. The improvements are especially
significant for a class of instances inspired from the image restoration
problem in computer vision. The magnitude of this effect is rather surprising
in that the 2-links are in relatively small number (quadratic in the number of
terms of the objective function).[Beamer] A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition
http://hdl.handle.net/2268/196557
Title: [Beamer] A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition
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<br/>Author, co-author: Deliège, Adrien; Nicolay, Samuel
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<br/>Abstract: We introduce a new method based on wavelets (EWMD)
for decomposing a signal into quasi-periodic oscillating components with
smooth time-varying amplitudes. This method is inspired by both the
“classic” wavelet-based decomposition and the empirical mode decomposition
(EMD). We compare the reconstruction skills and the period
detection ability of the method with the well-established EMD on toys
examples and the ENSO climate index. It appears that the EWMD accurately
decomposes and reconstructs a given signal (with the same efficiency
as the EMD), it is better at detecting prescribed periods and is
less sensitive to noise. This work provides the first version of the EWMD.
Even though there is still room for improvement, it turns out that preliminary
results are highly promising.Mars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure
http://hdl.handle.net/2268/196556
Title: Mars Topography Investigated Through the Wavelet Leaders Method: a Multidimensional Study of its Fractal Structure
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<br/>Author, co-author: Deliège, Adrien; Kleyntssens, Thomas; Nicolay, Samuel
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<br/>Abstract: This work examines the scaling properties of Mars topography through a wavelet-based formalism. We conduct exhaustive one-dimensional (both longitudinal and latitudinal) and two-dimensional studies based on Mars Orbiter Laser Altimeter (MOLA) data using the multifractal formalism called Wavelet Leaders Method (WLM). This approach shows that a scale break occurs at approximately 15 km, giving two scaling regimes in both 1D and 2D cases. At small scales, these topographic profiles mostly display a monofractal behavior while a switch to
multifractality is observed in several areas at larger scales. The scaling exponents extracted from this framework tend to be greater at small scales. In the 1D context, these observations are in agreement with previous works and thus suggest that the WLM is well-suited for examining scaling properties of topographic fields. Moreover, the 2D analysis is the first such complete study to our knowledge. It gives both a local and global insight on the scaling regimes of the surface of Mars and allows to exhibit the link between the scaling exponents and several famous features of the Martian topography. These results may be used as a solid basis for further investigations of the scaling laws of the Red planet and show that the WLM could be used to perform systematic analyses of the surface roughness of other celestial bodies.Some characterizations about Generalized Hölder-Zygmund Spaces Λ_{σ,N}^{α}(R^d)
http://hdl.handle.net/2268/196532
Title: Some characterizations about Generalized Hölder-Zygmund Spaces Λ_{σ,N}^{α}(R^d)
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<br/>Author, co-author: Kreit, Damien; Nicolay, Samuel
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<br/>Abstract: Generalized Hölder-Zygmund spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ were recently introduced and are based on a generalization of Besov spaces. Under some conditions, generalized Hölder-Zygmund and Besov spaces are equal. It has been proved that most properties of classical Hölder-Zygmund spaces are held for spaces $\Lambda^{\sigma,\alpha}(\R^{d})$, which constitute a particular case of spaces $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ with $N_{j}=2^{j}$. The goal of the present document is to prove that most of these properties are kept for $\Lambda_{\sigma, N}^{\alpha}(\R^{d})$ spaces.Sur la question 482
http://hdl.handle.net/2268/196518
Title: Sur la question 482
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<br/>Author, co-author: Catalan, EugèneSur les branches infinies des courbes algébriques (suite)
http://hdl.handle.net/2268/196517
Title: Sur les branches infinies des courbes algébriques (suite)
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<br/>Author, co-author: Catalan, EugèneSur les branches infinies des courbes algébriques
http://hdl.handle.net/2268/196516
Title: Sur les branches infinies des courbes algébriques
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<br/>Author, co-author: Catalan, Eugène