A study of singular integral operators with shift

Nesta tese, consideram-se operadores integrais singulares com a acção extra de um operador de deslocacamento de Carleman e com coeficientes em diferentes classes de funções essencialmente limitadas. Nomeadamente, funções contínuas por troços, funções quase-periódicas e funções possuíndo factorização generalizada. Nos casos dos operadores integrais singulares com deslocamento dado pelo operador de reflexão ou pelo operador de salto no círculo unitário complexo, obtêm-se critérios para a propriedade de Fredholm. Para os coeficientes contínuos, uma fórmula do índice de Fredholm é apresentada. Estes resultados são consequência das relações de equivalência explícitas entre aqueles operadores e alguns operadores adicionais, tais como o operador integral singular, operadores de Toeplitz e operadores de Toeplitz mais Hankel. Além disso, as relações de equivalência permitem-nos obter um critério de invertibilidade e fórmulas para os inversos laterais dos operadores iniciais com coeficientes factorizáveis. Adicionalmente, aplicamos técnicas de análise numérica, tais como métodos de colocação de polinómios, para o estudo da dimensão do núcleo dos dois tipos de operadores integrais singulares com coeficientes contínuos por troços. Esta abordagem permite também a computação do inverso no sentido Moore-Penrose dos operadores principais. Para operadores integrais singulares com operadores de deslocamento do tipo Carleman preservando a orientação e com funções contínuas como coeficientes, são obtidos limites superiores da dimensão do núcleo. Tal é implementado utilizando algumas estimativas e com a ajuda de relações (explícitas) de equivalência entre operadores. Focamos ainda a nossa atenção na resolução e nas soluções de uma classe de equações integrais singulares com deslocamento que não pode ser reduzida a um problema de valor de fronteira binomial. De forma a atingir os objectivos propostos, foram utilizadas projecções complementares e identidades entre operadores. Desta forma, as equações em estudo são associadas a sistemas de equações integrais singulares. Estes sistemas são depois analisados utilizando um problema de valor de fronteira de Riemann. Este procedimento tem como consequência a construção das soluções das equações iniciais a partir das soluções de problemas de valor de fronteira de Riemann. Motivados por uma grande diversidade de aplicações, estendemos a definição de operador integral de Cauchy para espaços de Lebesgue sobre grupos topológicos. Assim, são investigadas as condições de invertibilidade dos operadores integrais neste contexto.

Regularity of Wiener-Hopf plus Hankel operators

In this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols in the class of almost periodic, semi-almost periodic and piecewise almost periodic functions. In the first place, we consider Wiener-Hopf-Hankel operators acting between L2 Lebesgue spaces with possibly different Fourier matrix symbols in the Wiener-Hopf and in the Hankel operators. In the second place, we consider these operators with equal Fourier symbols and acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1 and w belongs to a subclass of Muckenhoupt weights. In addition, singular integral operators with Carleman shift and almost periodic coefficients are also object of study. The main purpose of this thesis is to obtain regularity properties characterizations of those classes of operators. By regularity properties we mean those that depend on the kernel and cokernel of the operator. The main techniques used are the equivalence relations between operators and the factorization theory. An invertibility characterization for the Wiener-Hopf-Hankel operators with symbols belonging to the Wiener subclass of almost periodic functions APW is obtained, assuming that a particular matrix function admits a numerical range bounded away from zero and based on the values of a certain mean motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly different AP symbols, criteria for the semi-Fredholm property and for one-sided and both-sided invertibility are obtained and the inverses for all possible cases are exhibited. For such results, a new type of AP factorization is introduced. Singular integral operators with Carleman shift and scalar almost periodic coefficients are also studied. Considering an auxiliar and simpler operator, and using appropriate factorizations, the dimensions of the kernels and cokernels of those operators are obtained. For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP matrix symbols and acting between L2-spaces, criteria for the Fredholm property are presented as well as the sum of the Fredholm indices of the Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. By studying dependencies between different matrix Fourier symbols of Wiener-Hopf plus Hankel operators acting between L2-spaces, results about the kernel and cokernel of those operators are derived. For Wiener-Hopf-Hankel operators acting between weighted Lebesgue spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols in the Wiener-Hopf and in the Hankel operators and belonging to the classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization for Wiener-Hopf plus Hankel operators with APp;w symbols. In the cases for which the Fourier symbols of the operators belong to SAPp;w and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel operators as well as formulas for the Fredholm indices of those operators.

Topology via enriched categories

Having as a starting point the characterization of probabilistic metric spaces as enriched categories over the quantale , conditions that allow the generalization of results relating Cauchy sequences, convergence of sequences, adjunctions of V-distributors and its representability are established. Equivalence between L-completeness and L-injectivity is also established. L-completeness is characterized via the Yoneda embedding, and injectivity is related with exponentiability. Another kind of completeness is considered and the formal ball model is analyzed.

Multilayered micro/nanocrystalline CVD diamond coatings for biotribology

In the present work multilayered micro/nanocrystalline (MCD/NCD) diamond coatings were developed by Hot Filament Chemical Vapour Deposition (HFCVD). The aim was to minimize the surface roughness with a top NCD layer, to maximize adhesion onto the Si3N4 ceramic substrates with a starting MCD coating and to improve the mechanical resistance by the presence of MCD/NCD interfaces in these composite coatings. This set of features assures high wear resistance and low friction coefficients which, combined to diamond biocompatibility, set this material as ideal for biotribological applications. The deposition parameters of MCD were optimized using the Taguchi method, and two varieties of NCD were used: NCD-1, grown in a methane rich gas phase, and NCD-2 where a third gas, Argon, was added to the gas mixture. The best combination of surface pre-treatments in the Si3N4 substrates is obtained by polishing the substrates with a 15 μm diamond slurry, further dry etching with CF4 plasma for 10 minutes and final ultrasonic seeding in a diamond powder suspension in ethanol for 1 hour. The interfaces of the multilayered CVD diamond films were characterized with high detail using HRTEM, STEM-EDX and EELS. The results show that at the transition from MCD to NCD a thin precursor graphitic film is formed. On the contrary, the transition of the NCD to MCD grade is free of carbon structures other than diamond, as a result of the richer atomic hydrogen content and of the higher substrate temperature for MCD deposition. At those transitions, WC nanoparticles were found due to contamination from the filament, being also present at the first interface of the MCD layer with the silicon nitride substrate. In order to study the adhesion and mechanical resistance of the diamond coatings, indentation and particle jet blasting tests were conducted, as well as tribological experiments with homologous pairs. Indentation tests proved the superior behaviour of the multilayered coatings that attained a load of 800 N without delamination, when compared to the mono and bilayered ones. The multilayered diamond coatings also reveal the best solid particle erosion resistance, due to the MCD/NCD interfaces that act as crack deflectors. These results were confirmed by an analytical model on the stress field distribution based on the von Mises criterion. Regarding the tribological testing under dry sliding, multilayered coatings also exhibit the highest critical load values (200N for Multilayers with NCD-2). Low friction coefficient values in the range μ=0.02- 0.09 and wear coefficient values in the order of ~10-7 mm3 N-1 m-1 were obtained for the ball and flat specimens indicating a mild wear regime. Under lubrication with physiological fluids (HBSS e FBS), lower wear coefficient values ~10-9-10-8 mm3 N-1 m-1) were achieved, governed by the initial surface roughness and the effective contact pressure.

14º Encontro Internacional de Arte e Tecnologia: #14.ART: arte e desenvolvimento humano

#14ART: Arte e Desenvolvimento Humano propõe-se discutir novos territórios para uma maior sustentabilidade, assim como debater futuras evoluções criativas. O evento procurará entrepor-se em zonas de contato entre domínios tradicionalmente separadas - a arte e a ciência, pesquisa acadêmica e práticas criativas independentes, politicas sustentáveis e engajamento social, para o século XXI. Pretende-se que a discussão se centre, sobretudo, sobre como explorar o potencial transformativo da arte na pós-média. Hoje, de acordo com vários pensadores - Rosalind Krauss, a Lev Manovich, Peter Weibel – estamos numa fase pós- média; não existe um só meio, nos nossos dias, que domine o discurso da pratica artística contemporânea no campo dos média, bem pelo contrário os média encontram-se, hoje, engajados no pensamento critico do discurso da contemporaneidade. Através dos eventos anteriores deste Encontro Internacional, ficou claro que a arte hoje - com as condições proporcionadas pelo discurso do pós-média - oferece um potencial muito mais inteligente e interessante elocução para as artes. No entanto, as qualidade simbólicas e estéticas, bem como o pensamento critico e os aspectos investigativos e de confronto teórico da “pré-média arte”, também se apresentam ser tão importantes para a “pós-media arte”, obrigando o discurso artístico a manter um fio condutor entre o físico (a obra) e o mental (conceito) - realidades e utopias.