Let 𝑋 be a compact subset of the complex plane and denote by 𝑅𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿𝑝(𝑋) norm. We show that if a point 𝑥0 admits a bounded point derivation on 𝑅𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥0. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if a point 𝑥0 admits a bounded point derivation on 𝑅(𝑋) and if 𝑋 contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to 𝑥0. We also extend this result to the case of higher order bounded point derivations. These results were first shown by O'Farrell; however, we prove them constructively by explicitly using the Cauchy integral formula.

No description available.

In this thesis, we aim to study derivations from l^1(Z+) to its dual, l^\infty(Z+). We first characterize them as certain closed subspace of l^1(Z+). Then we present a necessary and sufficient condition, due to M. J. Heath, to make a bounded derivation on l^1(Z+) into l^\infty(Z+), a compact linear operator. After that base on the work in [6], we study weakly compact derivations from l^1(Z+) to its dual. We introduce T-sets and TF-sets and then state their relation with weakly compact operators on l^1(Z+). These results are originally due to Y. Choi and M. J. Heath, but we give simpler proofs. Finally, we will study certain classes of derivations from L^1(R+) to L^\infty(R+), and give an elementary proof that they are always mapped into C0(R+).

The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The mode of production used as raw material for (re) production of natural resources, as different as possible. It is, therefore justified the removal of vegetation, for real estate, agropastoral and mining, causing serious damage to the environment and the contamination of river channels through the dumping of sewage in natura. The coastal town Estância/Sergipe no exception to this reality, where production and occupancy of their spaces not meeting the timely rebalancing of environmental systems is maintaining environmental sustainability. The city is located east of the state of Sergipe, with a territory of 647.62 km² and is part of the south coast. With a proposal for a geoenvironmental analysis of the municipality aimed to identify the natural physical environments and anthropogenic lead currently processed in resort and its possible consequences to the local environment. The analysis method was based on geo approach, suitable for integrated analysis of various environmental units contained in the municipality and the methodological procedures were based on geospatial technologies, preparation of thematic maps, field work, reading relevant documents to the investigation as bibliographies consistent with the research object and the applied theory, knowledge of the socioeconomic and environmental processes in the city to present, from the integrated analysis, to identify the environmental damage in the landscape units from human activity. The municipality of Estância space is a very dynamic nature, consisting of fragile ecosystems to man's actions. Patches of Atlantic Forest and mangrove ecosystem is periodically cleared, burned and filled to make way for housing and economic activities. The waters of the river channels that bathe the city are polluted mainly by domestic sewage and fecal coliform. Thus, it becomes necessary to develop research projects that assist in the formulation of public policies, planning, use and preservation of natural spaces, since the balance of the physical environment is fundamental to the welfare and maintenance of man on the surface of planet Earth. / O modo de produção utiliza como matéria-prima para sua (re)produção os recursos naturais, os mais diversos possíveis. Assim, é justificada a retirada da cobertura vegetal, para fins imobiliários, agropastoris e de extração mineral, causando sérios danos ao meio ambiente, bem como a contaminação dos canais fluviais através do despejo do esgoto doméstico in natura. O município costeiro de Estância/Sergipe não foge a esta realidade, onde a produção e ocupação dos seus espaços não respeitam o tempo hábil dos sistemas ambientais reequilibrarem-se, mantendo a sustentabilidade ambiental. O município localiza-se a leste do estado de Sergipe, com um território de 647,62km² e faz parte do litoral sul do estado. Com uma proposta de análise geoambiental do município, objetivou-se identificar os ambientes físicos naturais e as derivações antrópicas processadas atualmente em Estância e suas possíveis consequências ao meio ambiente local. O método de análise baseou-se na abordagem geossistêmica, apropriada para análises integradas das diversas unidades ambientais contidas no município e os procedimentos metodológicos basearam-se em técnicas de geoprocessamento, elaboração de cartas temáticas, trabalhos de campo, leituras de documentos pertinentes à investigação como bibliografias condizentes com o objeto de pesquisa e com a teoria aplicada, conhecimento dos processos socioeconômicos e ambientais presentes no município para, a partir da análise integrada, identificar os danos ambientais, nas unidades de paisagem, provenientes da ação antrópica. O território do município de Estância é um espaço físico natural muito dinâmico, constituído por ecossistemas frágeis às ações do homem. Manchas de Mata Atlântica e o ecossistema manguezal são periodicamente desmatados, queimados e aterrados para dar lugar a moradias e as atividades econômicas. As águas dos canais fluviais que banham o município estão contaminadas, principalmente por esgotos domésticos e coliformes fecais. Assim, torna-se necessário o desenvolvimento de projetos de pesquisa que auxiliem na formulação de políticas públicas, no ordenamento, no uso e na preservação dos espaços naturais, uma vez que o equilíbrio do meio físico é fundamental para o bem estar e manutenção do homem na superfície do planeta Terra.

Seja d:= \'\\partial\'/\'\\partial IND.x\'+ \'beta\\partial\'/\'partial IND.y\'uma derivação simples de K[x,y], onde K é um corpo de característica zero. Doering, Lequain e Ripoll ([1]) provaram que exite um \'gama\'\'PERTENCE A\' K[x,y] tal que o operador S = \'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'+\'gama\'\'PERTENCE A\'\'A IND.2\'\':= K[x,y]\' < \'\\partial\'/partial IND.x\', \'\\partial\'/\'partial\'/\'partial IND y\'\'>\'gera um ideal à esquerda maximal principal de \'A IND.2\'. Desta maneira mostraram, para n=2, que a seguinte conjectura é verdadeira: Seja d:=\'\\partial\'/ \'\\partial IND.x\"IND.1\"+\"alfa\'IND.2\'\'\\partial\'/\'\\partial\'IND.x\'\'IND.2\"+...+ alfa IND.n\"\\partial\'/\'\\partial IND.x\'\'IND.n\" uma derivaçào simples de K[\'x IND.1\'...\'x IND n\']. Então, A IND.n\'(d+\'gama\') é um ideal à esquerda maximal principal de Á IND.n\', para algum \'gama\'\'PERTENCE A\'K[\'x IND.1\',...\'x IND.n\']. Nós mostramos que esta conjectura é verdadeira em alguns casos particulares / Let d: =\'\\partial/\'/\'\\partial IND.x\'+ \'beta\\partial IND.y\' be a simple derivation of K[x,y], where K is a field of characteristic zero. Doering, Lequain e Ripol ([1]) proved that there exists a polynomial um \'gama\'\'IT BELONGS\' K[x,y] such that the operador S =\'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'\'gama\'\'IT BELONGS\'\' á ind.2\':= K[x,y]\' < \'\\partial\'/\'partial IND.x\',\'partial\'/\'partial\'/\'partial IND y\'> \'generates a principal maximal left ideal of A IND.2\'. In this way, they showed that, for n=2, the following conjectures is tru: Let d:=\'\\partial\'/\'\\partial IND.x\"+\"alfaÍND.2\"\\partial\'/ \"\\partial\' IND.x\'IND.2\"+ álfa IND.n\"\\partial IND.xÍND.n\"be a simple derivation of K[\'x IND.1\',...,\'x IND n\']. Then, \'A IND.n\'(d+\'gama\') is a principal maximal left ideal of \'A IND.n\',for some \'gama\"IT BELONGS\'K[x IND.1\',...,\'x IND.n\']. We show that this conjecture is true in some cases

Seja d:= \'\\partial\'/\'\\partial IND.x\'+ \'beta\\partial\'/\'partial IND.y\'uma derivação simples de K[x,y], onde K é um corpo de característica zero. Doering, Lequain e Ripoll ([1]) provaram que exite um \'gama\'\'PERTENCE A\' K[x,y] tal que o operador S = \'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'+\'gama\'\'PERTENCE A\'\'A IND.2\'\':= K[x,y]\' < \'\\partial\'/partial IND.x\', \'\\partial\'/\'partial\'/\'partial IND y\'\'>\'gera um ideal à esquerda maximal principal de \'A IND.2\'. Desta maneira mostraram, para n=2, que a seguinte conjectura é verdadeira: Seja d:=\'\\partial\'/ \'\\partial IND.x\"IND.1\"+\"alfa\'IND.2\'\'\\partial\'/\'\\partial\'IND.x\'\'IND.2\"+...+ alfa IND.n\"\\partial\'/\'\\partial IND.x\'\'IND.n\" uma derivaçào simples de K[\'x IND.1\'...\'x IND n\']. Então, A IND.n\'(d+\'gama\') é um ideal à esquerda maximal principal de Á IND.n\', para algum \'gama\'\'PERTENCE A\'K[\'x IND.1\',...\'x IND.n\']. Nós mostramos que esta conjectura é verdadeira em alguns casos particulares / Let d: =\'\\partial/\'/\'\\partial IND.x\'+ \'beta\\partial IND.y\' be a simple derivation of K[x,y], where K is a field of characteristic zero. Doering, Lequain e Ripol ([1]) proved that there exists a polynomial um \'gama\'\'IT BELONGS\' K[x,y] such that the operador S =\'\\partial\'/\'\\partial x\'+\'beta\\partial\'/\'\\partial y\'\'gama\'\'IT BELONGS\'\' á ind.2\':= K[x,y]\' < \'\\partial\'/\'partial IND.x\',\'partial\'/\'partial\'/\'partial IND y\'> \'generates a principal maximal left ideal of A IND.2\'. In this way, they showed that, for n=2, the following conjectures is tru: Let d:=\'\\partial\'/\'\\partial IND.x\"+\"alfaÍND.2\"\\partial\'/ \"\\partial\' IND.x\'IND.2\"+ álfa IND.n\"\\partial IND.xÍND.n\"be a simple derivation of K[\'x IND.1\',...,\'x IND n\']. Then, \'A IND.n\'(d+\'gama\') is a principal maximal left ideal of \'A IND.n\',for some \'gama\"IT BELONGS\'K[x IND.1\',...,\'x IND.n\']. We show that this conjecture is true in some cases

Let $\bk$ be a field of characteristic zero. A Pham-Brieskorn ring is a $\bk$-algebra of the form $B_{a_0,\dots,a_n} = \bk[X_0,\dots,X_n] / \lb X_0^{a_0} + \cdots + X_n^{a_n} \rb$, where $n \geq 2$ and $a_0, \dots, a_n$ are positive integers. A ring $B$ is rigid if the only locally nilpotent derivation $D : B \to B$ is the zero derivation. Consider the following conjecture. \begin{conjnonumber}\label{PBConjectureAbstract} Let $n \geq 2$, and let $B_{a_0, \dots, a_n} = \bk[X_0, \dots, X_n] / \langle X_0^{a_0} + \cdots + X_n^{a_n} \rangle$ be a Pham-Brieskorn ring. If $\min\{a_0, \dots,a_n \} \geq 2$ and at most one element $i$ of $\{0,\dots ,n\}$ satisfies $a_i = 2$, then $B_{a_0, \dots, a_n}$ is rigid. \end{conjnonumber} The $n = 2$ case of the Conjecture is known to be true. In this thesis, we make progress towards solving the above conjecture. Our main results are: \begin{enumerate}[\rm(1)] \item For any $n \geq 3$, in order to prove the above conjecture, it suffices to prove rigidity of $B_{a_0, \dots, a_n}$ in the cases where $\bk = \Comp$ and $\cotype(a_0, \dots, a_n) = 0$. \item For any $n \geq 2$, $X = \Proj B_{a_0, \dots, a_n}$ is a well-formed quasismooth weighted complete intersection if and only if $\cotype(a_0, \dots, a_n) = 0$. \item When $n = 3$ and $\cotype(a_0, a_1, a_2, a_3) = 0$, $B_{a_0, a_1, a_2, a_3}$ is rigid, except possibly in the cases where, up to a permutation of the $a_i$, $(a_0, a_1, a_2, a_3) \in \{(2,3,4,12), (2,3,5,30)\}$. \item We summarize the list of 3-dimensional Pham-Brieskorn rings $B_{a_0, a_1, a_2, a_3}$ for which rigidity is known. It follows in particular that if $B_{2,3,4,12}$ and $B_{2,3,5,30}$ are rigid then the $n = 3$ case of the above conjecture is true. \end{enumerate} In addition to the above, we develop techniques for proving rigidity of rings in general; prove rigidity of many Pham-Brieskorn rings whose dimension is greater than 3; give simple examples of rational projective surfaces with quotient singularities that have an ample canonical divisor and prove that the members of a certain family of singular hypersurfaces are not rational.

No description available.

No description available.