ORBi Collection: Mathematics
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LaTeX, un peu, beaucoup (7. Tableaux (encore))
http://hdl.handle.net/2268/179732
Title: LaTeX, un peu, beaucoup (7. Tableaux (encore))
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<br/>Author, co-author: Dupont, PascalLaTeX, un peu, beaucoup (6. Tableaux)
http://hdl.handle.net/2268/179731
Title: LaTeX, un peu, beaucoup (6. Tableaux)
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<br/>Author, co-author: Dupont, PascalLaTeX, un peu, beaucoup (5. Les macros)
http://hdl.handle.net/2268/179730
Title: LaTeX, un peu, beaucoup (5. Les macros)
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<br/>Author, co-author: Dupont, PascalÉvolutions et initiatives contribuant à faire vivre les maths comme une discipline vivante accessible à tous : le cas de la Belgique
http://hdl.handle.net/2268/179627
Title: Évolutions et initiatives contribuant à faire vivre les maths comme une discipline vivante accessible à tous : le cas de la Belgique
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<br/>Author, co-author: Henry, ValérieAngles corniculaires et de demi-cercle chez Euclide
http://hdl.handle.net/2268/179515
Title: Angles corniculaires et de demi-cercle chez Euclide
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<br/>Author, co-author: Bair, Jacques; Henry, Valérie
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<br/>Abstract: Dans cet article, on examine comment Euclide présentait, dans ses Eléments, les notions liées au concept d'angle dans le plan. Il considérait des angles mixtilignes, en particulier corniculaires et de demi-cercle, qui peuvent être de nos jours exploités pour introduire les nombres hyperréels intervenant en analyse non standard.Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.
http://hdl.handle.net/2268/179471
Title: Graded-commutative nonassociative algebras: higher octonions and Krichever-Novikov superalgebras; their structures, combinatorics and non-trivial cocycles.
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<br/>Author, co-author: Kreusch, Marie
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<br/>Abstract: This dissertation consists of two parts.
The first one is the study of a series of real (resp. complex) noncommutative and nonassociative algebras $\bbO_{p,q}$ (resp. $\bbO_{n}$) generalizing the algebra of octonion numbers $\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras.
Introduced by Morier-Genoud and Ovsienko, these algebras have a natural $\bbZ_2^n$-grading ($p+q =n$), and they are characterized by a cubic form over the field $\bbZ_2.$
We establish all the possible isomorphisms between the algebras $\bbO_{p,q}$
preserving the structure of $\bbZ_2^n$-graded algebra.
The classification table of $\bbO_{p,q}$ is quite similar to that of
the real Clifford algebras $\cC l_{p,q}$,
the main difference is that the algebras $\bbO_{n,0}$ and $\bbO_{0,n}$ are exceptional.
We also provide a periodicity for the algebras $\bbO_n$ and $\bbO_{p,q}$ analogous to the periodicity for the Clifford algebras $\cC l_{n}$ and $\cC l_{p,q}$. In the second part we consider superalgebras of Krichever-Novikov (K-N) type.
Krichever and Novikov introduced a family of Lie algebras with two marked points generalizing the Witt algebra and its central extension called the Virasoro algebra. The K-N Lie (super)algebras for more than two marked points were studied by Schlichenmaier.
In particular, he extended the explicit formula of $2$-cocycles due to Krichever and Novikov to multiple-point situation.
We give an explicit construction of central extensions of Lie superalgebras of K-N type and we establish a $1$-cocycle with values in its dual space.
In the case of Jordan superalgebras related to superalgebras of K-N type, we calculate a 1-cocycle with coefficients in the dual space.A comparison of within-season yield prediction algorithms based on crop model behaviour analysis
http://hdl.handle.net/2268/178919
Title: A comparison of within-season yield prediction algorithms based on crop model behaviour analysis
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<br/>Author, co-author: Dumont, Benjamin; Basso, Bruno; Leemans, Vincent; Bodson, Bernard; Destain, Jean-Pierre; Destain, Marie-France
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<br/>Abstract: The development of methodologies for predicting crop yield, in real-time and in response to different agro-climatic conditions, could help to improve the farm management decision process by providing an analysis of expected yields in relation to the costs of investment in particular practices. Based on the use of crop models, this paper compares the ability of two methodologies to predict wheat yield (Triticum aestivum L.), one based on stochastically generated climatic data and the other on mean climate data. It was shown that the numerical experimental yield distribution could be considered as a log-normal distribution. This function is representative of the overall model behaviour. The lack of statistical differences between the numerical realisations and the logistic curve showed in turn that the Generalised Central Limit Theorem (GCLT) was applicable to our case study. In addition, the predictions obtained using both climatic inputs were found to be similar at the inter and intra-annual time-steps, with the root mean square and normalised deviation values below an acceptable level of 10% in 90% of the climatic situations. The predictive observed lead-times were also similar for both approaches. Given (i) the mathematical formulation of crop models, (ii) the applicability of the CLT and GLTC to the climatic inputs and model outputs, respectively, and (iii) the equivalence of the predictive abilities, it could be concluded that the two methodologies were equally valid in terms of yield prediction. These observations indicated that the Convergence in Law Theorem was applicable in this case study. For purely predictive purposes, the findings favoured an algorithm based on a mean climate approach, which needed far less time (by 300-fold) to run and converge on same predictive lead time than the stochastic approach.Special issue dedicated to the 14th "Journées montoises d'informatique théorique"
http://hdl.handle.net/2268/178709
Title: Special issue dedicated to the 14th "Journées montoises d'informatique théorique"
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<br/>Editor: Bruyère, Véronique; Jungers, Raphaël; Hollanders; Rigo, Michel
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<br/>Commentary: The conference was held in Louvain-La-Neuve, September 11th - 14th, 2012Special issue dedicated to the second "AutoMathA conference"
http://hdl.handle.net/2268/178707
Title: Special issue dedicated to the second "AutoMathA conference"
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<br/>Editor: Bruyère, Véronique; Pin, Jean-Eric; Restivo, Antonio; Rigo, Michel
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<br/>Commentary: The conference was held in Liège June 8-12, 2009.Defining multiplication in some additive expansions of polynomial rings
http://hdl.handle.net/2268/178705
Title: Defining multiplication in some additive expansions of polynomial rings
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<br/>Author, co-author: Point, Françoise; Rigo, Michel; Waxweiler, Laurent
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<br/>Abstract: Adapting a result of R. Villemaire on expansions of Presburger arithmetic, we show how to define multiplication in some expansions of the additive reduct of certain Euclidean rings. In particular, this applies to polynomial rings over a finite field.Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words (long version)
http://hdl.handle.net/2268/178703
Title: Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words (long version)
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<br/>Author, co-author: Rigo, Michel; Salimov, Pavel
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<br/>Abstract: The binomial coefficient of two words $u$ and $v$ is the number of times $v$ occurs as a subsequence of $u$. Based on this classical notion, we introduce the $m$-binomial equivalence of two words refining the abelian equivalence. Two words $x$ and $y$ are $m$-binomially equivalent, if, for all words $v$ of length at most $m$, the binomial coefficients of $x$ and $v$ and respectively, $y$ and $v$ are equal. The $m$-binomial complexity of an infinite word $x$ maps an integer $n$ to the number of $m$-binomial equivalence classes of factors of length $n$
occurring in $x$. We study the first properties of $m$-binomial equivalence. We compute the $m$-binomial complexity of two classes of words: Sturmian words and (pure) morphic words that are fixed points of Parikh-constant morphisms like the Thue--Morse word, i.e., images by the morphism of all the letters have the same Parikh vector. We prove that the frequency of each symbol of an infinite recurrent word with bounded $2$-binomial complexity is rational.
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<br/>Commentary: This is an extended version of the conference version. In particular, it contains a new discussion about frequencies of symbols when the $2$-binomial complexity is bounded.Avoiding 2-binomial squares and cubes
http://hdl.handle.net/2268/178702
Title: Avoiding 2-binomial squares and cubes
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<br/>Author, co-author: Rao, Michaël; Rigo, Michel; Salimov, Pavel
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<br/>Abstract: Two finite words $u,v$ are $2$-binomially equivalent if, for all words $x$ of length at most $2$, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement of the usual abelian equivalence. A $2$-binomial square is a word $uv$ where $u$ and $v$ are $2$-binomially equivalent.
In this paper, considering pure morphic words, we prove that $2$-binomial squares (resp. cubes) are avoidable over a $3$-letter (resp. $2$-letter) alphabet. The sizes of the alphabets are optimal.On the number of abelian bordered words (with an example of automatic theorem-proving)
http://hdl.handle.net/2268/178701
Title: On the number of abelian bordered words (with an example of automatic theorem-proving)
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<br/>Author, co-author: Goc, Daniel; Rampersad, Narad; Rigo, Michel; Salimov, Pavel
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<br/>Abstract: In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in $\mathbb{Z}$ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a $3$-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue--Morse word using some kind of automatic theorem-proving provided by a logical characterization of the $k$-automatic sequences.
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<br/>Commentary: This is an extended version of the conference paper.A New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-Automatic Sequences
http://hdl.handle.net/2268/178495
Title: A New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-Automatic Sequences
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<br/>Author, co-author: Parreau, Aline; Rigo, Michel; Rowland, Eric; Vandomme, Elise
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<br/>Abstract: We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the l-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have 2-abelian complexity sequences that are 2-regular. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular.A Vectorial Regularization of the Keplerian Motion
http://hdl.handle.net/2268/178454
Title: A Vectorial Regularization of the Keplerian Motion
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<br/>Author, co-author: Condurache, D.; Martinusi, VladimirAbout the Rectilinear Keplerian Motion
http://hdl.handle.net/2268/178453
Title: About the Rectilinear Keplerian Motion
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<br/>Author, co-author: Condurache, D.; Martinusi, VladimirComputing the Logarithm of Homogenous Matrices in SE(3)
http://hdl.handle.net/2268/178452
Title: Computing the Logarithm of Homogenous Matrices in SE(3)
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<br/>Author, co-author: Condurache, D.; Martinusi, VladimirA Tensorial Explicit Solution to The Darboux Equation
http://hdl.handle.net/2268/178450
Title: A Tensorial Explicit Solution to The Darboux Equation
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<br/>Author, co-author: Condurache, D.; Martinusi, VladimirRemarks on the Hamiltonian of A Particle in A Rotating Reference Frame
http://hdl.handle.net/2268/178374
Title: Remarks on the Hamiltonian of A Particle in A Rotating Reference Frame
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<br/>Author, co-author: Martinusi, Vladimir; Condurache, D.Spectral properties of a pseudo-integrable map: the general case
http://hdl.handle.net/2268/178316
Title: Spectral properties of a pseudo-integrable map: the general case
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<br/>Author, co-author: Bogomolny, E; Dubertrand, Rémy; Schmit, C.
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<br/>Abstract: It is well established numerically that spectral statistics of pseudo-integrable models differs considerably from the reference statistics of integrable and chaotic systems. In Bogomolny and Schmit (2004 Phys. Rev. Lett. 93 254102) statistical properties of a certain quantized pseudo-integrable map had been calculated analytically but only for a special sequence of matrix dimensions.
The purpose of this paper is to obtain the spectral statistics of the same quantum map for all matrix dimensions.