The purpose of my research was to develop and strengthen my personality in music as a vocalist, so I would be able to freely let it out during any musical encounter. My base for the research was a duet with guitarist Mikael Máni Ásmundsson, but it was during a rehearsal with him that I realised for the first time I even had a musical personality – that I liked. Before, I had often felt uncomfortable and tense when playing music and not sure who I was as an artist. I limited myself to working with West Coast jazz, jazz standards, and lyrical improvisation, and I used various methods such as transcribing, ear-training in multiple forms, and improvisation exercises, to develop my musical personality. The research resulted in a nine-song album that Mikael Máni and I created in 2020 under the artist-name Tendra. That album is my examination project. I learned a great deal from the experience. I gained self-confidence, independence trust and calmness during this journey, and at last I feel comfortable with who I am as an artist – my musical personality has finally moved in.

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<p>We investigate the problem of the maximum number of distinct primitively rooted squares in a string. In comparison to considering general strings, the number of distinct symbols in the string is introduced as an additional parameter of the problem. Let S(d,n) = max {s(x) | x is a (d,n)-string}, where s(x) denotes the number of distinct primitively rooted squares in a string x and a (d,n)-string denotes a string of length n with exactly d distinct symbols.</p> <p>Inspired by the d-step approach which was instrumental in Santos' tackling of the Hirsch conjecture, we introduce a (d,n-d) table with entries S(d,n) where d is the index for the rows and n-d is the index for the columns. We examine the properties of the S(d,n) function in the context of (d,n-d) table and conjecture that the value of S(d,n) is no more than n-d. We present several equivalent properties with the conjecture. We discuss the significance of the main diagonal of the (d,n-d) table, i.e. the square-maximal (d, 2d)-strings for their relevance to the conjectured bound for all strings. We explore their structural properties under both assumptions, complying or not complying with the conjecture, with the intention to derive a contradiction. The result yields novel properties and statements equivalent with the conjecture with computational application to the determination of the values S(d,n).</p> <p>To further populate the (d,n-d) table, we design and implement an efficient computational framework for computing S(d,n). Instead of generating all possible (d,n)-strings as the brute-force approach needs to do, the computational effort is significantly reduced by narrowing down the search space for square-maximal strings. With an easily accessible lower bound obtained either from the previously computed values inductively or by an effective heuristic search, only a relatively small set of candidate strings that might possibly exceed the lower bound is generated. To this end, the notions of s-cover and the density of a string are introduced and utilized. In special circumstances, the computational efficiency can be further improved by starting the s-cover with a double square structure. In addition, we present an auxiliary algorithm that returns the required information including the number of distinct squares for each generated candidate string. This algorithm is a modified version of FJW algorithm, an implementation based on Crochemore's partition algorithm, developed by Franek, Jiang and Weng. As of writing of this thesis, we have been able to obtain the maximum number of distinct squares in binary strings till the length of 70.</p> / Doctor of Philosophy (PhD)

O artigo Weighted forms of Euler's theorem de William Y.C. Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes ímpares e partes distintas via a introdução do conceito de partição com raiz. A propositura deste trabalho é envolta à apresentação de resultados sobre partições com raiz de modo a posteriormente realizar formulações combinatórias das identidades de Ramanujan por meio deste conceito, procurando estabelecer conexões com formas ponderadas do Teorema de Euler. Em particular, a bijeção de Sylvester e a iteração de Pak da função de Dyson são elementos primordiais para obtê-las. / The article Weighted forms of Euler's theorem by William Y.C. Chen and Kathy Q. Ji in response to the questioning of George E. Andrews, American mathematician, about nding combinatorial proofs for two identities in Ramanujan's Lost Notebook shows us some weighted forms of Euler's Theorem on partitions with odd parts and distinct parts through the introduction of the concept of rooted partition. The purpose of this work involves the presentation of results on rooted partitions in order to make combinatorial formulations of Ramanujan's identities, seeking to establish connections with weighted forms of Euler's Theorem. In particular, the Sylvester's bijection and the Pak's iteration of the Dyson's map are primordial elements to obtain them.

O artigo Weighted forms of Euler's theorem de William Y.C. Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes ímpares e partes distintas via a introdução do conceito de partição com raiz. A propositura deste trabalho é envolta à apresentação de resultados sobre partições com raiz de modo a posteriormente realizar formulações combinatórias das identidades de Ramanujan por meio deste conceito, procurando estabelecer conexões com formas ponderadas do Teorema de Euler. Em particular, a bijeção de Sylvester e a iteração de Pak da função de Dyson são elementos primordiais para obtê-las. / The article Weighted forms of Euler's theorem by William Y.C. Chen and Kathy Q. Ji in response to the questioning of George E. Andrews, American mathematician, about nding combinatorial proofs for two identities in Ramanujan's Lost Notebook shows us some weighted forms of Euler's Theorem on partitions with odd parts and distinct parts through the introduction of the concept of rooted partition. The purpose of this work involves the presentation of results on rooted partitions in order to make combinatorial formulations of Ramanujan's identities, seeking to establish connections with weighted forms of Euler's Theorem. In particular, the Sylvester's bijection and the Pak's iteration of the Dyson's map are primordial elements to obtain them.

Étant donnés un graphe simple non orienté G = (V, E) et un sommet particulier r dans V appelé racine, un arbre enraciné, ou r-arbre, de G est soit le graphe nul soit un arbre contenant r. Si un vecteur de capacités sur les sommets est donné, un sous-graphe de G est dit borné si le degré de chaque sommet dans le sous-graphe est inférieur ou égal à sa capacité. Soit w un vecteur de poids sur les arêtes et p un vecteur de profits sur les sommets. Le problème du r-arbre borné maximum (MBrT, de l’anglais Maximum Bounded r-Tree) consiste à trouver un r-arbre borné T = (U, F) de G tel que son poids soit maximisé. Si la contrainte de capacité du problème MBrT est relâchée, nous obtenons le problème du r-arbre maximum (MrT, de l’anglais Maximum r-Tree). Cette thèse contribue à l’étude des problèmes MBrT et MrT.Tout d’abord, ces deux problèmes sont formellement définis et leur complexité est étudiée. Nous présentons ensuite des polytopes associés ainsi qu’une formulation pour chacun d’entre eux. Par la suite, nous proposons plusieurs algorithmes combinatoires pour résoudre le problème MBrT (et donc le problème MrT) en temps polynomial sur les arbres, les cycles et les cactus. En particulier, un algorithme de programmation dynamique est utilisé pour résoudre le problème MBrT sur les arbres. Pour les cycles, nous sommes amenés a considérer trois cas différents pour lesquels le problem MBrT se réduit à certains problèmes polynomiaux. Pour les cactus, nous montrons tout d’abord que le problème MBrT peut être résolu en temps polynomial sur un type de graphes appelé cactus basis. En utilisant une série de décompositions en sous-problèmes sur les arbres et les cactus basis, nous obtenons un algorithme pour les graphes de type cactus.La deuxième partie de ce travail étudie la structure polyédrale de trois polytopes associés aux problèmes MBrT et MrT. Les deux premiers polytopes, Bxy(G,r,c) et Bx(G,r,c) sont associés au problème MBrT. Tous deux considèrent des variables sur les arêtes de G, mais seuls Bxy(G,r,c) possède également des variables sur les sommets de G. Le troisième polytope, Rx(G,r), est associé au problème MrT et repose uniquement sur les variables sur les arêtes. Pour chacun de ces trois polytopes, nous étudions sa dimension, caractérisons certaines inégalités définissant des facettes, et présentons les moyens possibles de décomposition. Nous introduisons également de nouvelles familles de contraintes. L’ajout de ces contraintes nous permettent de caractériser ces trois polytopes dans plusieurs classes de graphes.Pour finir, nous étudions les problèmes de séparation pour toutes les inégalités que nous avons trouvées jusqu’ici. Des algorithmes polynomiaux de séparation sont présentés, et lorsqu’un problème de séparation est NP-difficile, nous donnons des heuristiques de séparation. Tous les résultats théoriques développés dans ce travail sont implémentés dans plusieurs algorithmes de coupes et branchements auxquels une matheuristique est également jointe pour générer rapidement des solutions réalisables. Des expérimentations intensives ont été menées via le logiciel CPLEX afin de comparer les formulations renforcées et originales. Les résultats obtenus montrent de manière convaincante la force des formulations renforcées. / Given a simple undirected graph G = (V, E) with a so-called root node r in V, a rooted tree, or an r-tree, of G is either the empty graph, or a tree containing r. If a node-capacity vector c is given, then a subgraph of G is said to be bounded if the degree of each node in the subgraph does not exceed its capacity. Let w be an edge-weight vector and p a node-price vector. The Maximum Bounded r-Tree (MBrT) problem consists of finding a bounded r-tree T = (U, F) of G such that its weight is maximized. If the capacity constraint from the MBrT problem is relaxed, we then obtain the Maximum r-Tree (MrT) problem. This dissertation contributes to the study of the MBrT problem and the MrT problem.First we introduce the problems with their definitions and complexities. We define the associated polytopes along with a formulation for each of them. We present several polynomial-time combinatorial algorithms for both the MBrT problem (and thus the MrT problem) on trees, cycles and cactus graphs. Particularly, a dynamic-programming-based algorithm is used to solve the MBrT problem on trees, whereas on cycles we reduce it to some polynomially solvable problems in three different cases. For cactus graphs, we first show that the MBrT problem can be solved in polynomial time on a so-called cactus basis, then break down the problem on any cactus graph into a series of subproblems on trees and on cactus basis.The second part of this work investigates the polyhedral structure of three polytopes associated with the MBrT problem and the MrT problem, namely Bxy(G, r, c), Bx(G, r, c) and Rx(G, r). Bxy(G, r, c) and Bx(G, r, c) are polytopes associated with the MBrT problem, where Bxy(G, r, c) considers both edge- and node-indexed variables and Bx(G, r, c) considers only edge-indexed variables. Rx(G, r) is the polytope associated with the MrT problem that only considers edge-indexed variables. For each of the three polytopes, we study their dimensions, facets as well as possible ways of decomposition. We introduce some newly discovered constraints for each polytope, and show that these new constraints allow us to characterize them on several graph classes. Specifically, we provide characterization for Bxy (G, r, c) on cactus graphs with the help of a decomposition through 1-sum. On the other hand, a TDI-system that characterizes Bx(G,r,c) is given in each case of trees and cycles. The characterization of Rx(G,r) on trees and cycles then follows as an immediate result.Finally, we discuss the separation problems for all the inequalities we have found so far, and present algorithms or cut-generation heuristics accordingly. A couple of branch-and-cut frameworks are implemented to solve the MBrT problem together with a greedy-based matheuristic. We compare the performances of the enhanced formulations with the original formulations through intensive computational test, where the results demonstrate convincingly the strength of the enhanced formulations.

Nous étudions dans cette thèse l’algèbre de Hopf H associée à l’opérade pré-Lie. L’espace des éléments primitifs du dual gradué est muni d’une structure pré-Lie à gauche notée ⊲ définie par l’insertion d’un arbre dans un autre. Nous retrouvons la relation de dérivation entre le produit pré-Lie ⊲ et le produit pré-Lie de greffe → sur les éléments primitifs du dual gradué de l’algèbre de Hopf de Connes Kreimer HCK. Nous mettons en évidence un coproduit sur le produit tensoriel H ⊗HCK, qui en fait une algèbre de Hopf dont le dual gradué est isomorphe à l’algèbre enveloppante du produit semi-direct des deux algèbres de Lie considérées. Nous montrons que l’espace engendré par les arbres enracinés qui ont au moins une arête, muni du produit d’insertion, est une algèbre pré-Lie (non libre) engendrée par deux éléments. Nous mettons en évidence deux familles de relations. De plus nous montrons un résultat similaire pour l’algèbre pré-Lie associée à l’opérade NAP. Finalement on introduit les opérades à débit constant et on montre que l’opérade pré-Lie s’obtient comme déformation de l’opérade NAP dans ce cadre. / We investigate in this thesis the Hopf algebra structure on the vector space H spanned by the rooted forests, associated with the pre-Lie operad. The space of primitive elements of the graded dual of this Hopf algebra is endowed with a left pre-Lie product denoted by ⊲, defined in terms of insertion of a tree inside another. In this thesis we retrieve the “derivation” relation between the pre-Lie structure ⊲ and the left pre-Lie product → on the space of primitive elements of the graded dual H0CK of the Connes-Kreimer Hopf algebra HCK, defined by grafting. We also exhibit a coproduct on the tensor product H⊗HCK, making it a Hopf algebra the graded dual of which is isomorphic to the enveloping algebra of the semidirect product of the two (pre-)Lie algebras considered. We prove that the span of the rooted trees with at least one edge endowed with the pre-Lie product ⊲ is generated by two elements. It is not free : we exhibit two families of relations. Moreover we prove a similar result for the pre-Lie algebra associated with the NAP operad. Finally, we introduce current preserving operads and prove that the pre-Lie operad can be obtained as a deformation of the NAP operad in this framework.

The forest product industry is keenly interested in extending the normal planting season, as well as in the comparative field performance of standard nursery bare-root seedlings and containerized rooted cuttings. The effect of seasonal planting dates on survival, above and belowground biomass allocation, water relations, gas exchange attributes and foliar carbon isotope composition (δ13C) of two stock types of slash pine (Pinus elliottii Engelm.) were examined. Slash pine bare-root seedlings (BRS) and containerized rooted cuttings (CRC) were hand planted in September, November, January, March and April in three consecutive planting seasons (2000-2001, 2001-2002 and 2002-2003) on three sites with silt loam topsoils in southwestern Louisiana. First-year mean survival of CRC across all planting dates and sites was consistently high at 96 to 98%, whereas BRS survival was significantly (P < 0.0001) lower at 59 to 81% and highly variable among study sites and dates through three planting seasons. Generally, there was a negative relationship between soil moisture at the time of planting and first-year survival of BRS planted September through March in 2001-2002 and 2002-2003 planting seasons, whereas the opposite was observed only for BRS planted in April 2002 and 2003. Survival of CRC was affected very little by the variation in soil moisture. Containerized rooted cuttings had higher early above and belowground biomass, and height and diameter than did BRS. However, three years after planting the size differences between stock types disappeared or became negligible. Early size differences among trees planted September through March also decreased after three years, although September trees were tallest. Growth of the April-planted trees was poor compared to trees planted in other months. Late-planted April trees had higher δ13C values, and higher water-use efficiency in the first growing season compared to earlier planted trees. Differences in δ13C values among the planting dates disappeared in the second growing season. Net photosynthesis rates did not differ considerably between stock types or among planting dates in the second and third growing seasons. This study indicates that it is possible to extend the planting season to as early as September and as late as March by using CRC.

The forest product industry is keenly interested in extending the normal planting season, as well as in the comparative field performance of standard nursery bare-root seedlings and containerized rooted cuttings. The effect of seasonal planting dates on survival, above and belowground biomass allocation, water relations, gas exchange attributes and foliar carbon isotope composition (&#948;13C) of two stock types of slash pine (Pinus elliottii Engelm.) were examined. Slash pine bare-root seedlings (BRS) and containerized rooted cuttings (CRC) were hand planted in September, November, January, March and April in three consecutive planting seasons (2000-2001, 2001-2002 and 2002-2003) on three sites with silt loam topsoils in southwestern Louisiana. First-year mean survival of CRC across all planting dates and sites was consistently high at 96 to 98%, whereas BRS survival was significantly (P < 0.0001) lower at 59 to 81% and highly variable among study sites and dates through three planting seasons. Generally, there was a negative relationship between soil moisture at the time of planting and first-year survival of BRS planted September through March in 2001-2002 and 2002-2003 planting seasons, whereas the opposite was observed only for BRS planted in April 2002 and 2003. Survival of CRC was affected very little by the variation in soil moisture. Containerized rooted cuttings had higher early above and belowground biomass, and height and diameter than did BRS. However, three years after planting the size differences between stock types disappeared or became negligible. Early size differences among trees planted September through March also decreased after three years, although September trees were tallest. Growth of the April-planted trees was poor compared to trees planted in other months. Late-planted April trees had higher &#948;13C values, and higher water-use efficiency in the first growing season compared to earlier planted trees. Differences in &#948;13C values among the planting dates disappeared in the second growing season. Net photosynthesis rates did not differ considerably between stock types or among planting dates in the second and third growing seasons. This study indicates that it is possible to extend the planting season to as early as September and as late as March by using CRC.

Dans cette thèse, nous étudions le concept d’algèbre pré-Lie libre engendrée par un ensemble (non-vide). Nous rappelons la construction par A. Agrachev et R. Gamkrelidze des bases de monômes dans les algèbres pré-Lie libres. Nous décrivons la matrice des vecteurs d’une base de monômes en termes de la base d’arbres enracinés exposée par F. Chapoton et M. Livernet. Nous montrons que cette matrice est unipotente et trouvons une expression explicite pour les coefficients de cette matrice, en adaptant une procédure suggérée par K. Ebrahimi-Fard et D. Manchon pour l’algèbre magmatique libre. Nous construisons une structure d’algèbre pré-Lie sur l’algèbre de Lie libre $\mathcal{L}$(E) engendrée par un ensemble E, donnant une présentation explicite de $\mathcal{L}$(E) comme quotient de l’algèbre pré-Lie libre $\mathcal{T}$^E, engendrée par les arbres enracinés (non-planaires) E-décorés, par un certain idéal I. Nous étudions les bases de Gröbner pour les algèbres de Lie libres dans une présentation à l’aide d’arbres. Nous décomposons la base d’arbres enracinés planaires E-décorés en deux parties O(J) et $\mathcal{T}$(J), où J est l’idéal définissant $\mathcal{L}$(E) comme quotient de l’algèbre magmatique libre engendrée par E. Ici, $\mathcal{T}$(J) est l’ensemble des termes maximaux des éléments de J, et son complément O(J) définit alors une base de $\mathcal{L}$(E). Nous obtenons un des résultats importants de cette thèse (Théorème 3.12) sur la description de l’ensemble O(J) en termes d’arbres. Nous décrivons des bases de monômes pour l’algèbre pré-Lie (respectivement l’algèbre de Lie libre) $\mathcal{L}$(E), en utilisant les procédures de bases de Gröbner et la base de monômes pour l’algèbre pré-Lie libre obtenue dans le Chapitre 2. Enfin, nous étudions les développements de Magnus classique et pré-Lie, discutant comment nous pouvons trouver une formule de récurrence pour le cas pré-Lie qui intègre déjà l’identité pré-Lie. Nous donnons une vision combinatoire d’une méthode numérique proposée par S. Blanes, F. Casas, et J. Ros, sur une écriture du développement de Magnus classique, utilisant la structure pré-Lie de $\mathcal{L}$(E). / In this thesis, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction by A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by F. Chapoton and M. Livernet. Also, we show that this matrix is unipotent and we find an explicit expression for its coefficients, adapting a procedure implemented for the free magmatic algebra by K. Ebrahimi-Fard and D. Manchon. We construct a pre-Lie structure on the free Lie algebra $\mathcal{L}$(E) generated by a set E, giving an explicit presentation of $\mathcal{L}$(E) as the quotient of the free pre-Lie algebra $\mathcal{T}$^E, generated by the (non-planar) E-decorated rooted trees, by some ideal I. We study the Gröbner bases for free Lie algebras in tree version. We split the basis of E- decorated planar rooted trees into two parts O(J) and $\mathcal{T}$(J), where J is the ideal defining $\mathcal{L}$(E) as a quotient of the free magmatic algebra generated by E. Here $\mathcal{T}$(J) is the set of maximal terms of elements of J, and its complement O(J) then defines a basis of $\mathcal{L}$(E). We get one of the important results in this thesis (Theorem 3.12), on the description of the set O(J) in terms of trees. We describe monomial bases for the pre-Lie (respectively free Lie) algebra $\mathcal{L}$(E), using the procedure of Gröbner bases and the monomial basis for the free pre-Lie algebra obtained in Chapter 2. Finally, we study the so-called classical and pre-Lie Magnus expansions, discussing how we can find a recursion for the pre-Lie case which already incorporates the pre-Lie identity. We give a combinatorial vision of a numerical method proposed by S. Blanes, F. Casas, and J. Ros, on a writing of the classical Magnus expansion in $\mathcal{L}$(E), using the pre-Lie structure.