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壹、引言 近年來，訊號傳送的途徑，已擺脫了傳統上著重管線傳送的優勢；有愈來愈多的訊號 彌漫在廣闊的空間裡，而這種無線式的傳送所需面臨的問題是：不具有排它性，任何 有接收器材的非原始接收者都可以截聽到訊息，由於因應而生的保密技術格外受矚目 ，密碼學（CRYPTOGRAPHY）便是滿足此需要的學問。本論文所探討的排列多項式（PE RMUTATION POLYNOMIAL）是密碼學中重要的工具之一。 貳、論文主體 所謂排列多項式，即是佈於代數體上的多項式，把此多項式當成函數而作用於代數體 （FIELD ）上，如果此函數具有一對一的性質，則是排列多項式。即 ｆ（ｘ）＝ａ。+ ａ1 ｘ1 + ....ａnｘn ≡ Ｆq〔Ｘ〕且 ｆ（ａ）╪ｆ（ｂ），ａ，ｂ≡Ｆｑ，ａ╪ｂ． 在論文中，介紹先進學者對排列多項式的認識。如：LAGRANGE'S INTERPOLATION是利 用函數值來描繪多項式，著名的學者CARLITZ ，利用特殊多項式來合成出排列多項式 ，論文中有更進一步的合成法提出，而HERMITE 跟DICKSON 學者則提出Ft函數其冪次 的變化情形，來判別排列多項式之是否，是最通俗的判別理論。 此外，由吾人所蒐集的資料中發現，在祗有兩項的多項式中，被發現到其它更簡捷快 速的判別方法，故二項式的多項式的探討是本論文的第一主題，對於 k j Ｘ＋ｂｘ ≡Ｆｑ〔Ｘ〕， 給予固定類型的ｑ，ｋ，ｊ情形下，祗須檢定ｂ是否具特 殊性質就可決定是否為排列多項式，這是一種方法。另有學者並不固定ｑ，ｋ，ｊ， 反而從ｑ，ｋ，ｊ數字下手，找尋出某種關連性，其結果使得係數ｂ，只有當ｂ＝０ ，時才有機會是排列多項式，乘下單項式的判別過程，就很容易了。另外還有一種方 法也是找尋ｑ，ｋ，ｊ間的關係，不過其結果在找出：多項式為非排列多項式，是比 較特別的地方。上述三方法，本論文網羅大部份有關論文，綜合各家之長，並適當給 予一同於原作者的新觀點證明方法。 至於本論文第二主題是著名的CARLITZ'S CONJECTURE此預測敘述：對於任何具有最高 冪次是偶數的多項式，必定存在一個自然數ｋ，使得給定的代數體，其元素個數只要 超過ｋ，則此多項式必定不是排列多項式。此預測當degree n=10,12,14, and 2m 時 已被證實為真。本論文僅就n=2m，做系統地探討及重新證明。 參、結語 本論文所論的兩主題，對於佈於代數體上的多項式是否為排列多項式，在判別的過程 上應有相當的助益才是。

In this thesis, we focus on the following topics: supereulerian graphs, hamiltonian line graphs, fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees, and several extremal problems on the (minimum and/or maximum) size of graphs under a given graph property. The thesis includes six chapters. The first one is to introduce definitions and summary the main results of the thesis, and in the last chapter we introduce the furture research of the thesis. The main studies in Chapters 2 - 5 are as follows. In Chapter 2, we explore conditions for a graph to be supereulerian.In Section 1 of Chapter 2, we characterize the graphs with minimum degree at least 2 and matching number at most 3. By using the characterization, we strengthen the result in [93] and we also address a conjecture in the paper.In Section 2 of Chapter 2, we prove that if $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$, then $G$ is collapsible except for several special graphs, where $p(n)=0$ for $n$ even and $p(n)=1$ for $n$ odd. As a corollary, a characterization for graphs satisfying $d(x)+d(y)\geq n-1-p(n)$ for any edge $xy\in E(G)$ to be supereulerian is obtained. This result extends the result in [21].In Section 3 of Chapter 2, we focus on a conjecture posed by Chen and Lai [Conjecture~8.6 of [33]] that every 3-edge connected and essentially 6-edge connected graph is collapsible. We find a kind of sufficient conditions for a 3-edge connected graph to be collapsible.In Chapter 3, we mainly consider the hamiltonicity of 3-connected line graphs.In the first section of Chapter 3, we give several conditions for a line graph to be hamiltonian, especially we show that every 3-connected, essentially 11-connected line graph is hamilton- connected which strengthens the result in [91].In the second section of Chapter 3, we show that every 3-connected, essentially 10-connected line graph is hamiltonian-connected.In the third section of Chapter 3, we show that 3-connected, essentially 4-connected line graph of a graph with at most 9 vertices of degree 3 is hamiltonian. Moreover, if $G$ has 10 vertices of degree 3 and its line graph is not hamiltonian, then $G$ can be contractible to the Petersen graph.In Chapter 4, we consider edge fault-tolerant hamiltonicity of Cayley graphs generated by transposition trees. We first show that for any $F\subseteq E(Cay(B:S_{n}))$, if $|F|\leq n-3$ and $n\geq4$, then there exists a hamiltonian path in $Cay(B:S_{n})-F$ between every pair of vertices which are in different partite sets. Furthermore, we strengthen the above result in the second section by showing that $Cay(S_n,B)-F$ is bipancyclic if $Cay(S_n,B)$ is not a star graph, $n\geq4$ and $|F|\leq n-3$.In Chapter 5, we consider several extremal problems on the size of graphs.In Section 1 of Chapter 5, we bounds the size of the subgraph induced by $m$ vertices of hypercubes. We show that a subgraph induced by $m$ (denote $m$ by $\sum\limits_{i=0}^ {s}2^{t_i}$, $t_0=[\log_2m]$ and $t_i= [\log_2({m-\sum\limits_{r=0}^{i-1}2 ^{t_r}})]$ for $i\geq1$) vertices of an $n$-cube (hypercube) has at most $\sum\limits_{i=0}^{s}t_i2^{t_i-1} +\sum\limits_{i=0}^{s} i\cdot2^{t_i}$ edges. As its applications, we determine the $m$-extra edge-connectivity of hypercubes for $m\leq2^{[\frac{n}2]}$ and $g$-extra edge-connectivity of the folded hypercube for $g\leq n$.In Section 2 of Chapter 5, we partially study the minimum size of graphs with a given minimum degree and a given edge degree. As an application, we characterize some kinds of minimumrestricted edge connected graphs.In Section 3 of Chapter 5, we consider the minimum size of graphs satisfying Ore-condition.

A ênfase do trabalho recai na análise da taxa temporal de produção de entropia em sistemas de rede determinados por suas propriedades de simetria, no intuito de constituí-la como uma ferramenta paradigmática no estudo de sistemas irreversíveis. Nesse sentido, uma vez que se consegue consolidar uma definição consistente dessa grandeza para este campo, propõe-se uma abordagem estocástica para o modelamento da dinâmica dos exemplares considerados. O grupo de simetria C_ descreve as propriedades de invariância em sistemas com ampla relevância em física, de maneira que resulta natural elaborar a construção dos modelos em torno a sua estrutura, mais ainda quando se pode assumir que as características de simetria, no relativo à determinação da classe de universalidade dos modelos, são condições mais relevantes que a presença do não-equilíbrio, em coerência com a conjetura de Grinstein. Aliás, o modelo de Potts de três estados também apresenta propriedades de simetria próprias daquele grupo, além de ser passível de certo rigor no seu tratamento matemático, de maneira que oferece resultados consideravelmente satisfatórios e propícios para, por comparação, analisar os sistemas que concernem neste estudo. Assim, o procedimento tem como fim a determinação numérica da produção de entropia em sistemas com dinâmica irreversível e invariantes ante as transformações de simetria que compõem o grupo C_3v, partindo para isso de simulações de Monte Carlo em modelos estruturados sobre redes quadradas. A determinação da produção de entropia segue a prescrição de Schnakenberg (Schnakenberg [1976]), fundamentada nas correntes de probabilidade que surgem no sistema como consequência da violação da reversibilidade microscópica; a qual, por sua vez, estabelece a necessidade para os sistemas em equilíbrio de que todas as sequências cíclicas possíveis entre estados consecutivos sejam percorridas com igualdade de probabilidade num sentido quanto no inverso. Como uma segunda instância deste trabalho, também foram estudadas as propriedades de escalabilidade apresentadas por estos modelos durante seus primeiros instantes de evolução, isso, a partir da determinação numérica dos exponentes dinâmicos e sua caracterização dentro do marco teórico conhecido como \"short time scaling\'\' (Jansen et al. (1989)). Nesse sentido foram consideradas algumas das prescrições concernentes que tem apresentado melhor desempenho nos últimos anos. O observado mostra um comportamento coerente com a satisfação do que implicaria a conjetura de Grinstein para estes sistemas irreversíveis, indicando sua pertença à classe de universalidade do modelo de Potts de três estados, e, com isso, reafirmando os resultados obtidos em relação à produção de entropia. / This work is proposed as a study of the entropy production rate in lattice systems determined by its symmetries, looking for its consolidation as a paradigmatic tool in the area of irreversible and nonequilibrium systems. Henceforth, given the actual possibility of defining this quantity on that field, a stochastic perspective is adopted for modeling the dynamics of the considered systems. The C_ symmetry group describes the invariance properties of a wide range of physical systems, hence it results sensitive building the models to be dealt in accordance with its intrinsic characteristics, even more when it comes to be just fair, at the sight of the previous available analysis as well of the Grinstein conjecture, considering the invariance properties of systems as a more relevant factor than the reversibility conditions in what concerns the establishment of its universality class. The three states Potts model, indeed, shares its symmetry characteristics with those owned by the elements of the referred group and, also, concerning it there is a considerable amount of well confirmed information, becoming suitable for contrasting the results obtained. Henceforth, this work is focused on the numerical determination of the entropy production rate in irreversible systems whose invariance properties are the ones defined by the C_ symmetry group, implementing for that porpoise Monte Carlo simulations over square lattices models with the proper symmetries. By its own, the entropy production is determined in accordance with the Schnakenberg prescription (Scnhakenberg [1976]); deeply related with the probability currents emerging within irreversible systems as consequence of microscopic reversibility violation, which, in equilibrium, is imposed due to the mandatory equality between the evolution directions of all possible cyclic paths through a succession of states assumed by the system. As a second instance of this work, the scaling properties of the studied models during the first period of its evolution, just after the microscopic scale of time, were also analyzed. Henceforth, the determination of its dynamical exponents, as well as its characterization within the context of \"short time scaling\'\' (Jansent et al. [1989]) was realized through the calculus of some quantities with proven signatures of presenting an scalable \"initial slip\'\', finding strong suggestions for the models of being in the same universality class of the three state Potts model, fact coherent with the Grinstein conjecture if extended over them, but also with the observed behavior of the entropy production.

Submitted by Cláudia Bueno (claudiamoura18@gmail.com) on 2016-03-10T17:35:32Z No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-03-14T14:08:40Z (GMT) No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2016-03-14T14:08:40Z (GMT). No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-08-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based mainly on the Brunder and Godinho article [2] which shows proof of the conjecture of Artin methods using p-adic, although the conjecture is stated on the real numbers which makes the proof is show an equivalence on the field of the number p-adic method with the help of colored variables ya contraction of variables so as to prove the statement, taking the first level and ensuring a nontrivial solution in the following levels. / Este trabalho é baseado principalmente no artigo de Brunder e Godinho [2] o qual mostra a prova da conjetura de Artin usando métodos p-ádicos, ainda que a conjetura se afirma sobre o números reais o que faz a prova é mostrar uma equivalência sobre o corpo dos número p-ádicos com ajuda do método de variáveis coloridas e a contração de variáveis para assim provar a afirmação, tomando o primeiro nível e assim garantindo uma solução não trivial nos níveis seguintes.

Submitted by Luanna Matias (lua_matias@yahoo.com.br) on 2015-02-05T10:05:56Z No. of bitstreams: 2 Dissertaçao - Mónica Andrea Celis Cerón - 2014.pdf: 566862 bytes, checksum: b41da2ec2c63c537f6b78488d3d8c179 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-02-05T10:59:19Z (GMT) No. of bitstreams: 2 Dissertaçao - Mónica Andrea Celis Cerón - 2014.pdf: 566862 bytes, checksum: b41da2ec2c63c537f6b78488d3d8c179 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-02-05T10:59:19Z (GMT). No. of bitstreams: 2 Dissertaçao - Mónica Andrea Celis Cerón - 2014.pdf: 566862 bytes, checksum: b41da2ec2c63c537f6b78488d3d8c179 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-04-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Celis Cerón, Mónica Andrea. Artin’s conjecture for pairs of additive sextic forms. Goiânia, 2014. 62p. MSc. Dissertation. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Consider the system of equations a1xk1+ a2xk2+ + asxks= 0; b1xk1+ b2xk2+ + bsxks= 0; where a1; a2; ; as; b1; b2; ; bs 2 Z A special case of Artin’s conjecture states that the above system must have nontrivial solutions in every p-adic field, Qp, provided only that s 2k2+ 1. In this text we show that the conjecture is true when k = 6. / Celis Cerón, Mónica Andrea. Conjectura de Artin para pares de formas aditivas de grau 6. Goiânia, 2014. 62p. Dissertação de Mestrado. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Consideremos o sistema de equações a1xk1+ a2xk2+...+ asxks= 0; b1xk1+ b2xk2+ + bsxks= 0; onde, a 1; a 2; ; as; b1; b2; ; bs 2 Z. Um caso especial da conjectura de Artin nos diz que o sistema anterior tem solução não trivial em todo corpo p-ádico, Qp, sempre que s 2k2+ 1. Neste trabalho mostraremos que a conjectura é válida quando k = 6.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-08-06T13:53:45Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao_Alaides_Ferreira.pdf: 363902 bytes, checksum: 97bfa5be0bee9a9b8c283a12f0c24a18 (MD5) / Made available in DSpace on 2014-08-06T13:53:45Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao_Alaides_Ferreira.pdf: 363902 bytes, checksum: 97bfa5be0bee9a9b8c283a12f0c24a18 (MD5) Previous issue date: 2009 / This text is above solvability in systems of two forms additive over p-adics fields: with of degree k and variables n > 4k at lesat p > 3k4 ; with of degree an k odd integer at least n > 6k+1 variables; and with of degree 5 and p > 101 for n ≥ 31 variables, and for all p with n ≥ 36 variables, with the possible exceptions of p = 5 and p = 11. / Este texto é sobre solubilidade no corpo dos p-ádicos de sistemas de duas formas aditivas: com grau k e variáveis n > 4k apartir de p > 3k4 ; com grau k ímpar apartir de n > 6k +1 variáveis; e de grau 5 com p > 101 para n ≥ 31 variáveis, e para todo p com n ≥ 36 variáveis, com exceções de p = 5 e p = 11.

Provamos que \'C POT. 1\' genericamente difeomorfismos simpléticos ou são Anosov ou possuem entropia topológica limitada por baixo pelo supremo sobre o menor expoente de Lyapunov positivo dos pontos periódicos hiperbólicos. Usando isto exibimos exemplos de difeomorfismos conservativos sobre superfícies que não são pontos de semicontinuidade superior para a entropia topológica. Provamos também que \'C POT. 1\' genericamente difeomorfismos simpléticos não Anosov não admitem extensões simbólicas. Mudando de assunto, Hayashi estendeu um resultado de Mañé, provando que todo difeomorfismo f que possui uma \'C POT. 1\' vizinhança U, onde todos os pontos periódicos de qualquer g \'PERTENCE A\' U são hiperbólicos, é de fato um difeomorfismo Axioma A. Aqui, provamos o resultado análogo a este no caso conservativo, e a partir deste é possível exibir uma demonstração de um fato \"folclore\", a conjectura de Palis no caso conservativo / We prove that a \'C POT.1\' generic symplectic diffeomorphism is either Anosov or the topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of the periodic points. By means of that we give examples of area preserving diffeomorphisms which are not point of upper semicontinuity of entropy function in \'C POT. 1\' topology. We also prove that \'C POT. 1\'- generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension. Changing of subject, Hayashi has extended a result of Mañé, proving that every diffeomorphism f which has a \'C POT. 1\'-neighborhood U, where all periodic points of any g \'IT BELONGS\' U are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario, and using it we prove a \"folklore\" fact, the Palis conjecture in this context

Texte synthétique de présentation des théories quantiques des champs dites "hyperboliques" , définies par l'auteur en collaboration avec R. Benedetti. Leur place en topologie quantique, et leurs relations avec la conjecture du volume et les invariants de Chern-Simons, sont développés.

On propose quatre contributions à l'étude des fonctions L de formes modulaires. La première montre que le Jacobien d'une courbe modulaire possède un facteur simple sur le corps des rationnels de grande dimension et de rang nul, et un facteur simple de grande dimension et de grand rang. La seconde établit la conjecture de densité de niveau 1 des petits zéros pour de nouvelles familles de fonctions L de formes modulaires. La troisième étudie la distribution de la valeur en 1 de la fonction L de carré symétrique d'une forme modulaire. La dernière établit, en collaboration avec F. Martin, un critère de détermination des formes modulaires par les valeurs spéciales de leurs fonctions L.