Made available in DSpace on 2014-06-12T18:28:47Z (GMT). No. of bitstreams: 2 arquivo646_1.pdf: 442286 bytes, checksum: 8143ec3c3a5b067f3662ed8fb12ba045 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2007 / Universidade Federal de Sergipe / A álgebra de Gauss associada à k-subálgebra de um anel polinomial k[t0; : : : ; td] gerado por um número finito de formas de mesmo grau corresponde ao anel de coordenadas homogêneo da imagem de Gauss de uma variedade projetiva uniracional sobre k. Focaremos o caso onde os geradores são monômios. Por caracterizar os menores da matriz jacobiana de um conjunto de monômios como certos n-produtos tornaremos mais concreta a natureza da álgebra de Gauss associada à subálgebra monomial correspondente. A versão reticulada destes n-produtos permite uma abordagem combinatória ao tema. Neste caminho, provaremos resultados já obtidos e estudaremos em detalhes a álgebra de Gauss associada ao conjunto dos monômios livre de quadrados de grau dois

The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.

Tout d’abord, mon sujet de recherche porte sur le cryptographie à clé publique, plus précisément la cryptographie basée sur la théorie des codes correcteurs d’erreurs. L’objectif principal de cette thèse est d’analyser la sécurité des systèmes de chiffrement. Pour cela j’étudie les propriétés structurelles des différentes familles de codes linéaires utilisées dans la pratique. Mon travail de recherche s’est orienté de maniéré naturelle, vers l’étude des deux dernières propositions de cryptosystèmes, plus exactement le schéma de McEliece à base des codes MDPC [MTSB13](moderate parity check codes) et des codes Polaires [SK14]. Dans le cas des codes MDPC on a mis en évidence une faiblesse importante au niveau des clés utilisées par les utilisateurs du système. En effet, on a proposé un algorithme très efficace qui permet de retrouver une clé privé à partir d’une clé publique. Ensuite on a compté le nombre des clés faibles et on a utilisé le problème d’équivalence de codes pour élargir le nombre de clés faibles. On a publié notre travail de recherche dans une conférence internationale en cryptographie [BDLO16]. Ensuite on a étudié les codes Polaires et leur application à la cryptographie à clé publique. Depuis leur découverte par E. Arikan [Arı09], les codes Polaires font partie des famille de codes les plus étudié du point de vue de le théorie de l’information. Ce sont des codes très efficaces en terme de performance car ils atteignent la capacité des canaux binaires symétriques et ils admettent des algorithmes d’encodage et décodage très rapides. Néanmoins, peu des choses sont connu sur leur propriétés structurelles. Dans ce cadre la, on a introduit un formalisme algébrique qui nous a permit de révéler unegrande partie de la structure de ces codes. En effet, on a réussi à répondre à des questions fondamentales concernant les codes Polaires comme : le dual ou la distance minimale d’un code Polaire, le groupe des permutations ou le nombre des mots de poids faible d’un code Polaire. On a publié nos résultats dans une conférence internationale en théorie de l’information [BDOT16]. Par la suite on a réussi à faire une cryptanalyse complète du schéma de McEliece à base des codes Polaires. Ce résultat a été une application directe des propriétés découvertes sur les codes Polaires et il a été publié dans une conférence internationale en cryptographie post-quantique [BCD+16]. / First of all, during my PhD I focused on the public key cryptography, more exactly on the code-based cryptography. The main motivation is to study the security of the latest encryption schemes. For that, I analyzed in detail the structural properties of the main code families. Thus, my research was naturally directed to the study of the McEliece based encryption schemes, among which the latest MDCP based variant [MTSB13] and Polar codes variant [SK14]. In the case of the MDPC based variant, we manage to reveal an important weakness regarding the key pairs that are used in the protocol. Indeed, we proposed an efficient algorithm that retrieves the private key given the public key of the scheme. Next we counted the proportion of weak keys and we used the code equivalence problem to extend the number of weak keys. We published our results in an international conference in cryptography [BDLO16]. Next we studied the Polar codes and their application to public key cryptography.Since they were discovered by Arikan [Arı09], Polar codes are part of the most studied from an information theory point of view, family of codes. In terms of performance they are really efficient since they are capacity achieving over the Binary Discrete Memoryless Channels and they allow extremely fast encoding and decoding algorithms. Nonetheless, few facts are known about their structure. In this context, we have introduced an algebraic formalism which allowed us to reveal a big part of the structure of Polar codes. Indeed, we have managed to answer fundamental questions regarding Polar codes such as the dual, the minimum distance, the permutation group and the number of minimum weight codewords of a Polar code. Our results were published in an international conference in information theory [BDOT16]. We also managed to completely cryptanalyze the McEliece variant using Polar codes. The attack was a direct application of the aforementioned results on the structural properties of Polar codes and it was published in an international conference in postquantum cryptography [BCD+16].

The theory of Newton-Okounkov bodies can be viewed as a generalization of the theory of toric varieties; it associates a convex body to an arbitrary variety (equipped with auxiliary data). Although initial steps have been taken for formulating geometric situations under which the Newton-Okounkov body is a rational polytope, there is much that is still unknown. In particular, very few concrete and explicit examples have been computed thus far. In this thesis, we explicitly compute Newton-Okounkov bodies of some cases of Bott-Samelson and Peterson varieties (for certain classes of auxiliary data on these varieties). Both of these varieties arise, for instance, in the geometric study of representation theory. Background on the theory of Newton-Okounkov bodies and the geometry of flag and Grassmannian varieties is provided, and well as background on Bott-Samelson varieties, Hessenberg varieties, and Peterson varieties. In the last chapter we also discuss how certain techniques developed in this thesis can be generalized. In particular, a generalization of the flat family of Hessenberg varieties constructed in Chapter 6, which may allow us to compute Newton-Okounkov bodies of more general Peterson varieties, is an ongoing collaboration with H. Abe and M. Harada. / Thesis / Doctor of Philosophy (PhD)

This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP). In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig. / <p>QC 20180220</p>

In this thesis, we will give our discovery of original symmetric presentations of several important groups. We have investigated permutation and monomial progenitors 2*8: (23: 22), 2*9: (32: 24), 2*10: (24: (2 × 5)), 5*4:m (23: 22), 7*8:m (32: 24), and 3*5:m (24: (2 × 5)). The finite images of the above progenitors include the Mathieu sporadic group M12, the linear groups L2(8) and L2(13), and the extensions S6 × 2, 28 : .L2(8) , and 27 : .A5. We will show our construction of the four groups S3 , L2(8), L2(13), and S6 × 2 over S3, 22, S3 : 2, and S5, by using the technique of double coset enumeration. We will also provide isomorphism types all of the groups that have appeared as finite homomorphic images. We will show that the group L2(8) does not satisfy the conditions of Iwasawas Lemma and that the group L2(13) is simple by Iwasawas Lemma. We give constructions of M22 × 2 and M22 as homomorphic images of the progenitor S6.

Ha Minh Lam et M. Morales ont introduit une classe d'idéaux binomiaux qui est une extension binomiale d'idéaux monomiaux libres de carrés.Étant donné I un idéal monomial quadratique de k[x] libre de carrés et J une somme d'idéaux de scroll de k[z] qui satisfont certaines conditions, nous définissons l'extension binomiale de I comme B=I+J. Le sujet de cette thèse est d'étudier le nombre p plus grand tel que les sizygies de B son linéaires jusqu'au pas p-1. Sous certaines conditions d'ordre imposées sur les facettes du complexe de Stanley-Reisner de I nous obtiendrons un ordre > pour les variables de l'anneau de polynomes k[z]. Ensuite nous prouvons pour un calcul des bases de Gröbner que l'idéal initial in(B), sous l'ordre lexicographique induit par l'ordre de variables >, est quadratique libre de carrés. Nous montrerons que B est régulier si et seulement si I est 2-régulier. Dans le cas géneral, lorsque I n'est pas 2-régulier nous trouverons une borne pour l'entier q maximal qui satisfait que les premier q-1 sizygies de B son linéaires. En outre, en supossant que J est un idéal torique et en imposant des conditions supplémentaires, nous trouveron une borne supérieure pour l'entier q maximal qui satisfait que les premier q-1 sizygies de B son linéaires. En imposant des conditions supplémentaires, nous prouverons que les deux bornes sont égaux. / Ha Minh Lam et M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let I be a square free monomial ideal of k[x] and J a sum of scroll ideals in k[z] with some extra conditions, we define the binomial extension of $I$ as $B=I+Jsubset sis$. The aim of this thesis is to study the biggest number p such that the syzygies of B are linear until the step p-1. Due to some order conditions given to the facets of the Stanley-Reisner complex of I we get an order > for the variables of the polynomial ring k[z]. By a calculation of the Gröbner basis of the ideal $B$ we obtain that the initial ideal in(B) is a square free monomial ideal. We will prove that B is 2-regular iff I is 2-regular. In the general case, wheter I is not 2-regular we will find a lower bound for the the maximal integer q which satisfies that the first q-1 sizygies of B are linear. On the other hand, wheter J is toric and supposing other conditions, we will find a upper bound for the integer q which satisfies that the first q-1 syzygies of B are linear. By given more conditions we will prove that the twobounds are equal.

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.

The task of this thesis is to describe the dynamics and evolution of street terminology in Litomeřice, with a focus on its motivation and semantic structure of word formation. The material will be obtained with excerpts from archival materials, urban plans, lists of streets and squares, records of the municipal council and other historical and newer sources. When classifying a material, which will allow an overview of the motivational circuits, which are involved in the development of urbanonyms and used word-formation processes, it will underpin Šmilauer's semantic classification of anoikonyms and Dejmek's word formation and semantic classification of urbanonyms. The Development of the street terminology will be analyzed after each time period, the definition is determined by the circumstances, which had an considerable impact in the studied urbanonymia. Quantitative changes in the use of each motivation circuit and word-formation process will be illustrated by graph, all partial findings will be incorporated into the final synthetic whole. An important part of the work will consist an alphabetical list of names of current Litoměřice streets, squares and other public spaces including the time of their formation, interpretation of the names and brief overview of all urbanonyms, which preceded the...

Hlavním zájmem diplomové práce je studium zobecněné nabla Laplaceové transformace na časových škálach a její jednoznačnosti, včetně důkazu jednoznačnosti a aplikace jednoznačnosti v zlomkovém kalkulu na časových škálach.