A credit default swap is a contract that ensures protection against losses occurring due to a default event of an certain entity. It is crucial to know how default should be modelled for valuation or estimating of credit derivatives. In this dissertation, we first review the structural approach for modelling credit risk. The model is an approach for assessing the credit risk of a firm by typifying the firms equity as a European call option on its assets, with the strike price (or exercise price) being the promised debt repayment at the maturity. The model can be used to determine the probability that the firm will default (default probability) and the Credit Spread. We second concentrate on the valuation of credit derivatives, in particular the Credit Default Swap (CDS) when the hazard rate (or even of default) is modelled as the Vasicek-type model. The other objective is, by using South African credit spread data on defaultable bonds to estimate parameters on CIR and Vasicek-type Hazard rate models such as stochastic differential equation models of term structure. The parameters are estimated numerically by the Moment Method. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

Student Number : 0214677F - PhD thesis - School of Camputational and Applied Mathematics - Faculty of Science / Many real-life problems are formulated as global optimization problems with continuous variables. These problems are in most cases nonsmooth, nonconvex and often simulation based, making gradient based methods impossible to be used to solve them. Therefore, ef#2;cient, reliable and derivative-free global optimization methods for solving such problems are needed. In this thesis, we focus on improving the ef#2;ciency and reliability of some global optimization methods. In particular, we concentrate on improving some population set-based methods for unconstrained global optimization, mainly through hybridization. Hybridization has widely been recognized to be one of the most attractive areas of unconstrained global optimization. Experiments have shown that through hybridization, new methods that inherit the strength of the original elements but not their weakness can be formed. We suggest a number of new hybridized population set-based methods based on differential evolution (de), controlled random search (crs2) and real coded genetic algorithm (ga). We propose #2;ve new versions of de. In the #2;rst version, we introduce a localization, called random localization, in the mutation phase of de. In the second version, we propose a localization in the acceptance phase of de. In the third version, we form a de hybrid algorithm by probabilistically combining the point generation scheme of crs2 with that of de in the de algorithm. The fourth and #2;fth versions are also de hybrids. These versions hybridize the mutation of de with the point generation rule of the electromagnetism-like (em) algorithm. We also propose #2;ve new versions of crs2. The #2;rst version modi#2;es the point generation scheme of crs2 by introducing a local mutation technique. In the second and third modi#2;cations, we probabilistically combine the point generation scheme of crs2 with the linear interpolation scheme of a trust-region based method. The fourth version is a crs hybrid that probabilistically combines the quadratic interpolation scheme with the linear interpolation scheme in crs2. In the #2;fth version, we form a crs2 hybrid algorithm by probabilistically combining the point generation scheme of crs2 with that of de in the crs2 algorithm. Finally, we propose #2;ve new versions of the real coded genetic algorithm (ga) with arithmetic crossover. In the #2;rst version of ga, we introduce a local technique. We propose, in the second version, an integrated crossover rule that generates two children at a time using two different crossover rules. We introduce a local technique in the second version to obtain the third version. The fourth and #2;fth versions are based on the probabilistic adaptation of crossover rules. The ef#2;ciency and reliability of the new methods are evaluated through numerical experiments using a large test suite of both simple and dif#2;cult problems from the literature. Results indicate that the new hybrids are much better than their original counterparts both in reliability and ef#2;ciency. Therefore, the new hybrids proposed in this study offer an alternative to many currently available stochastic algorithms for solving global optimization problems in which the gradient information is not readily available.

Cette étude porte sur le contrôle moléculaire de l'expansion cellulaire dans l'apex de la racine du peuplier. L'étude a été focalisée sur la régulation de l'expression des aquaporines TIP1s sous deux niveaux de déficit hydrique. Un cadre conceptuel, combinant analyse de la densité de transcrits à haute résolution spatiale et un formalisme de la mécanique des fluides, a été établi pour décrire la régulation de l'expression des gènes dans le temps et l'espace, le long de l'apex. Deux états de croissance contrastés ont été étudiés : un où la croissance de l'apex de la racine est rétablie (après 3 jours de stress modéré, 80 mosmol kg-1, 100 g L-1 de polyéthylène glycol PEG à 3500 g mol-1) et un où elle est réduite (après 3 jours de stress fort, 250 mosmol kg-1, 200 g L-1 de PEG). L'analyse cinématique a révélé une sensibilité différentielle de la vitesse d'allongement relatif (REGR) selon le niveau de stress et la position le long de l'apex. Au niveau moléculaire, nous montrons que lorsque la croissance est réduite, les pics de densité de transcrits des TIP1s sont plus forts et décalés dans les premiers millimètres de l'apex, zone où la croissance des cellules est maintenue. Cependant les niveaux d'induction/répression n'étaient jamais plus élevés que dans la condition témoin, soulignant qu'une forte densité de transcrits n'est pas forcément synonyme d'une forte induction de l'expression. Focalisé sur la combinaison de deux processus dynamiques, l'expansion cellulaire et l'expression des gènes, cette étude a mis en évidence que les conclusions tirées de l'analyse de ces processus sont influencées par la façon dont le temps, l'espace sont considérés / This study consideres the molecular control of cell expansion in poplar root apex. The study was focused on the regulation of the TIP1 aquaporins expression under two levels of water deficit. A conceptual framework combining transcript density analysis (quantitative PCR) at a high spatial resolution and a fluid mechanics formalism was established to describe the regulation of gene expression in time and space along the root apex. Two contrasting growth status were both studied: root growth rate is either restored (after three days of moderate stress, 80 mmol kg-1, 100 g L-1 polyethylene glycol PEG 3500 g mol-1) or root growth rate is reduced (after 3 days of high stress, 250 mosmol kg-1, 200 g L-1 PEG). Kinematic analysis revealed a differential sensitivity of the relative elemental growth rate (REGR) according to the stress level and to the coordinate along the apex. At the molecular level, we showed that growth reduction was associated with a shift of maximal densities of transcripts towards the first millimeters of the apex, where cell expansion was maintained. Meanwhile the induction/repression levels were never stronger than in the control condition underlying that a high transcript density does not mean a high transcriptional induction. Focused on the combination of two dynamic processes, cell expansion and gene expression, my thesis showed that the conclusions issued from the analyses of these processes are influenced by the way time, space and age are considered

Le présent mémoire analyse l'impact du contexte d'insolvabilité sur le devoir fiduciaire d'agir dans le meilleur intérêt de la compagnie, devoir imposer aux administrateurs de compagnies par la législation corporative canadienne. L'objectif du mémoire est de déterminer un standard de conduite à être adopté par l'administrateur d'une compagnie insolvable en vue de répondre à ce devoir fiduciaire. Dans un premier temps, comment peut-on définir ce que constitue le « meilleur intérêt de la compagnie» ? L'auteur en vient à la conclusion que l'intérêt de la compagnie est au carrefour d'une communauté d'intérêts lui étant sous-jacents. L'intérêt de la compagnie, bien qu'indépendant de ces intérêts sous-jacents, ne peut s'analyser en faisant abstraction de ces derniers. La jurisprudence et la doctrine récentes laissent entrevoir que l'impact du contexte d'insolvabilité se fait sentir sur la détermination de ces intérêts sous-jacents à celui de la compagnie susceptibles d'être affectés par la finalité poursuivie par la compagnie, finalité axée sur la maximisation des profits à partir des opérations de l'entreprise exploitée par la compagnie. Dans un contexte d'insolvabilité, le créancier, à l'instar de l'actionnaire dans un contexte de solvabilité, supporte le risque commercial résiduel et doit recevoir une attention appropriée par les administrateurs. Par conséquent, dans la détermination de ce que constitue le meilleur intérêt de la compagnie, l'administrateur ne peut, lorsque la compagnie est insolvable, faire abstraction de l'intérêt des créanciers. Ainsi, dans un deuxième temps, qui sont les véritables bénéficiaires du devoir fiduciaire d'agir dans le meilleur intérêt de la compagnie dans un contexte d'insolvabilité? L'auteur en vient à la conclusion que le créancier est un bénéficiaire indirect de ce devoir fiduciaire lorsque la compagnie est insolvable. Tout comme l'actionnaire dans un contexte de solvabilité, le créancier doit être en mesure d'intenter un recours de nature dérivée en vue d'obtenir réparation, pour et au nom de la compagnie. Le contexte d'insolvabilité fait naître, à l'endroit des administrateurs, une obligation de nature fiduciaire de prendre en considération l'intérêt des créanciers tout en permettant à ces derniers d'intenter un tel recours dérivé en vue d'obtenir réparation à la suite d'une violation du devoir fiduciaire d'agir dans le meilleur intérêt de la compagnie. En plus d'être soutenue par une revue de la législation, de la jurisprudence et de la doctrine canadiennes, cette conclusion s'appuie sur une revue de la législation, de la jurisprudence et de la doctrine de certains pays du Commonwealth (Angleterre, Australie et Nouvelle-Zélande) et des États-Unis, juridictions avec lesquelles le Canada entretient des relations privilégiés, historiquement ou économiquement. Finalement, que doit faire l'administrateur d'une compagnie insolvable en vue de répondre à ce devoir fiduciaire d'agir dans le meilleur intérêt de la compagnie? L'auteur arrive à la conclusion que cette obligation de prendre en considération l'intérêt du créancier dans un contexte d'insolvabilité se traduit par un exercice de conciliation entre les intérêts du créancier et ceux des actionnaires. Les paramètres de cet exercice de conciliation sont déterminés en fonction du scénario envisagé par les administrateurs face à la situation d'insolvabilité. Plus le scénario se rapproche d'une liquidation plus ou moins formelle des actifs tangibles et facilement dissociables de la compagnie, moins cet exercice en sera un de conciliation et plus l'intérêt du créancier devra recevoir une attention prépondérante. À l'opposé, plus le scénario en est un de restructuration fondée sur une relance de l'entreprise exploitée par la compagnie insolvable, plus l'intérêt de l'actionnaire devra recevoir une attention particulière. / The following thesis analyses the impact of a company's insolvency on the fiduciary duties of its directors and officers, as imposed by Canadian corporate law. More specifically it shall examine the repercussions of an insolvency on management's fiduciary duty to act in accordance with the company's best interests. The objective shall therefore be to assess the extent of such fiduciary duty in the context of an insolvency and to establish a guideline to be followed by directors and officers in view of complying with said duty. Firstly, what constitutes the "best interests of the company"? The author concludes that a company's interests are comprised of a community of underlying interests. Although the interests of the company remain independent, it cannot be determined without taking into account said underlying interests. Amongst this community of underlying interests, shareholders and creditors, members of such community that supports the financing of the company's operations, occupy a predominant place. Recent case law and doctrinal authorities have concluded that the impact of a company's insolvency does not affect the ultimate objective pursued by the company, i.e. the maximisation of profit, but rather affects the determination of the members of the community of underlying interests that shall be affected by the pursuit of such finality. As such, in a context of insolvency, creditors, similar to shareholders in a context of solvency, are the residual risk-bearers of the company's commercial expenditure. Consequently, in view of determining what constitutes the best interests of the company, when it is insolvent, directors and officers may not ignore the creditors' interests. Secondly, who are the real beneficiaries of the fiduciary duty to act in a company's best interests when it becomes insolvent? The author concludes that creditors are indirect beneficiaries of such fiduciary duty in a context of insolvency. Similarly to shareholders in a context of solvency, creditors must be entitled to institute a derivative claim against directors and officers in order to obtain, in the name and for the company, compensation for the violation of said fiduciary duty. Insolvency triggers the existence of an obligation, fiduciary in nature, to take into account the creditors' interests. As such, directors and officers are to consider the creditors' interests in exercising their fiduciary duty to act in strict compliance with the best interests of the company. This conclusion is based on a review of the Canadian statutory law, jurisprudence and doctrinal authorities. Furthermore, this conclusion is supported by a review of same from certain Commonwealth jurisdictions (England, Australia and New-Zealand) and from the United States of America, jurisdictions with whom Canada has a privileged relationship, historically and/or economically. Finally, how does the obligation to take into account the creditors' interests translate in practice for directors and officers of an insolvent company? The author concludes that the obligation to take into account the creditors' interests is, in reality, an obligation to reconcile the shareholders' and creditors' interests with that of the company's best interests. The parameters of such obligation shall be determined with respect to the scenario conceived by directors and officers in order to effectively manage a situation of insolvency. Should such scenario focus on the liquidation of the core assets of the company, then the interests of the shareholders shall be subjugated to that of the creditors'. Conversely, should such scenario focus on the reorganisation of the company's commercial expenditures, based on the company's existing structures, then the shareholder's interest shall receive a more extensive attention by directors and officers.

This thesis is concerned with the mathematical analysis of electricity and carbon emission markets. We introduce a novel, versatile and tractable stochastic framework for the joint price formation of electricity spot prices and allowance certificates. In the proposed framework electricity and allowance prices are explained as functions of specific fundamental factors, such as the demand for electricity and the prices of the fuels used for its production. As a result, the proposed model very clearly captures the complex dependency of the modelled prices on the aforementioned fundamental factors. The allowance price is obtained as the solution to a coupled forward-backward stochastic differential equation. We provide a rigorous proof of the existence and uniqueness of a solution to this equation and analyse its behaviour using asymptotic techniques. The essence of the model for the electricity price is a carefully chosen and explicitly constructed function representing the supply curve in the electricity market. The model we propose accommodates most regulatory features that are commonly found in implementations of emissions trading systems and we analyse in detail the impact these features have on the prices of allowance certificates. Thereby we reveal a weakness in existing regulatory frameworks, which, in rare cases, can lead to allowance prices that do not conform with the conditions imposed by the regulator. We illustrate the applicability of our model to the pricing of derivative contracts, in particular clean spread options and numerically illustrate its ability to "see" relationships between the fundamental variables and the option contract, which are usually unobserved by other commonly used models in the literature. The results we obtain constitute flexible tools that help to efficiently evaluate the financial impact current or future implementations of emissions trading systems have on participants in these markets.

The subject of this thesis is the real-time implementation of algebraic derivative estimators as observers in nonlinear control of magnetic levitation systems. These estimators are based on operational calculus and implemented as FIR filters, resulting on a feasible real-time implementation. The algebraic method provide a fast, non-asymptotic state estimation. For the magnetic levitation systems, the algebraic estimators may replace the standard asymptotic observers assuring very good performance and robustness. To validate the estimators as observers in closed-loop control, several nonlinear controllers are proposed and implemented in a experimental magnetic levitation prototype. The results show an excellent performance of the proposed control laws together with the algebraic estimators. / O tema dessa dissertação é a implementação em tempo real dos estimadores algébricos de derivadas como observadores no controle não-linear de levitação magnética. Esses estimadores são baseados no cálculo operacional e implementados como filtros FIR, resultando em uma implementação viável em tempo real. O método algébrico permite estimar os estados do sistema de maneira rápida e não-assintótica. Para os sistemas de levitação magnética, os estimadores algébricos podem substituir os observadores assintóticos assegurando boas propriedades de robustez e performance. A fim de validar os estimadores como observadores no controle em malha fechada, vários controladores não-lineares são propostos e implementados em um protótipo experimental. Os resultados mostram uma excelente performance dos controladores propostos juntamente com os estimadores algébricos.

O cetoconazol é um derivado imidazólico com atividade antifúngica, empregado no tratamento de grande variedade de infecções cutâneas. O objetivo deste trabalho foi a padronização e a validação dos métodos analíticos empregando-se a espectrofotometria no ultravioleta ( UV ) por derivada de primeira ordem e a cromatografia líquida de alta eficiência (CLAE) para a determinação do cetoconazol em formulações farmacêuticas sob a forma de creme obtidas no comércio e formuladas em laboratório. O estudo da estabilidade física das formulações obtidas em laboratório foi realizado após o armazenamento sob diferentes condições de tempo, temperatura e umidade. O método espectrofotométrico no UV por derivada de primeira ordem foi padronizado a 257nm , no intervalo de concentração de 5,0 a 30,0 µg/mL em metanol. O coeficiente de correlação linear foi de 0,9997, o percentual de recuperação foi de 100,24%. A média do desvio padrão relativo foi de 0,56% para a amostra simulada e 0,41% para a amostra comercial. A determinação de cetoconazol por CLAE foi realizada empregando-se uma coluna LiChrospher® 100 RP- 18 ( 5µm ) e fase móvel constituída por uma mistura de trietilamina em metanol( 1:500 ) e solução de acetato de amônio em água (1:200 ) na proporção de 75 : 25 v/v, com vazão de 1,0 mL/min e detecção no UV a 225 nm. O terconazol foi utilizado como padrão interno. O coeficiente de correlação linear foi de 0,9981, o percentual de recuperação foi de 100,88%, a média do desvio padrão foi de 2,13% para a amostra simulada e 1,25% para a amostra comercial. A amostra simulada apresentou comportamento pseudoplástico e tixotropia em decorrência das medidas reológicas obtidas experimentalmente. / Ketoconazole is an imidazole antifungal agent used in the treatment of great variety of skin infectious diseases. The aim of this research was the standardization and the validation of two analytical methods, first derivative ultraviolet ( UV ) spectrophotometry and high performance liquid chromatography( HPLC ) to determine ketoconazole in commercial and simulated cream dosage forms preparations. The validated methods were used to study the stability of the simulated cream after storage under high temperatures and high percentage of humidity. First derivative UV spectrophotometry was standardized at 257 nm, in a range of concentration from 5.0 to 30.0 µg/mL in methanol. The coefficient of correlation was 0.9997 and the recovery average was 100.24%. The average relative standard deviation ( RSD ) was 0.56% for simulated preparation and 0.41 % for commercial preparation. The determination of ketoconazole by HPLC was performed using a LiChrospher® 100 RP-18(5µm ) column, a mobile phase consisting of triethylamine in methanol (1:500) and ammonium acetate solution in water(1:200 ) 75:25 ( v/v ), a flow rate of 1.0 mL/min and UV detection at 225nm. Terconazole was used as internal standard. The coefficient of correlation was 0.9981, the recovery average was 100.88%, the average relative standard deviation ( RSD ) was 2.13% for the simulated preparations and 1.25% for the commercial preparations. The rheological measurements showed that the simulated preparation presented pseudoplastic and thixotropic behavior.

A prática da reutilização de produtos médico-hospitalares de uso único vem sendo aplicada desde meados da década de setenta. A principal razão que tem contribuído para disseminação desta conduta pelas instituições hospitalares radicadas tanto nos países em desenvolvimento como naqueles considerados ricos, tem sido a aparente economia de custos. Apesar dos riscos relacionados com a prática da reutilização, como reações pirogênicas, danos ocasionados por bactérias consideradas patogênicas em pacientes imunologicamente comprometidos, danos na integridade fisica dos produtos, assim como aumento do período de permanência dos pacientes no hospital, têm despertado o interesse em avaliar aspectos fisicos e biológicos dos produtos médico-hospitalares reutilizados. Baseando-se nestas considerações foram aplicados desafios com esporos de Bacillus Subtilis varo niger ATCC 9372 e endotoxina bacteriana E. coli 055:B5. Os produtos desafiados foram cateteres intravenosos, torneira três vias e tubos de traqueostomia. A possível presença microbiana foi investigada após contaminação intencional dos esporos de B. Subtillis (107 ufc/unid.) com submissão das unidades contaminadas à limpeza e posterior esterilização, utilizando óxido de etileno/CFC na proporção 12:88. Os ciclos de reprocessamentos simulados de produtos médico-hospitalares consistiram de contaminação de cada unidade teste com carga microbiana, lavagem com detergente enzimático, secagem e esterilização. Ao término de cada ciclo de reprocessamento foram separadas unidades representativas para avaliação por contagem microbiana (pour plate), testes de esterilidade por inoculação direta e indireta, citotoxidade por cultura de células e microscopia eletrônica de varredura. A eficiência da esterilidade foi avaliada tanto por contagem microbiana como pelos testes de esterilidade, que resultaram em níveis microbianos de 103 ufc/unid. e detecção de contaminação até o 6° ciclo de reprocessamento nos cateteres intravenosos, tubos de traqueostomia e torneiras três vias. A segurança dos reprocessamentos dos produtos médico-hospitalares foi avaliada pela cultura de células de fibroblastos de camundongo (NCTC clone 929), as quais não apresentaram toxicidade. Entretanto, os resultados obtidos durante microscopia eletrônica de varredura comprovaram presença de carga microbiana após 10° ciclo de reprocessamento, assim como danos na superficie polimérica. Durante desafio com endotoxina bacteriana, que consistiu em contaminar as unidades com 200 UE, secagem e exposição ao ciclo de esterilização com óxido de etileno/CFC (12:88), verificou-se que após ciclos de reprocessamentos simulados, totalizando dez ciclos, foi possível detectar valores de recuperação de endotoxina em torno de 100%. Os cateteres-guia que foram adquiridos em instituição hospitalar após quatro reutilizações, apresentaram níveis de contaminação de 105 ufc/unid., assim como presença de bactérias consideradas patogênicas em pacientes comprometidos imunologicamente, já a detecção de endotoxina bacteriana nestes cateteres não foi considerada significativa. Logo, as avaliações aplicadas nas unidades submetidas aos ciclos de reprocessamentos simulados, assim como nos cateteres-guia reprocessados e reutilizados quatro vezes, refletiram a realidade de algumas instituições no âmbito nacional e internacional que praticam a reutilização de produtos médico-hospitalares de uso-único. Os resultados obtidos vêm enfatizar objeções quanto à prática da reutilização, considerando que a ausência de segurança pode ocasionar em danos ao paciente. / The practices of reutilization of single use hospital medical devices came into practice since mid seventies. The main reason that has contributed for dissemination of this practice by institutional hospitals in underdeveloped as well as those considered as rich countries has been the appellant economy of costs. Despite well known risks related with the practice of reutilization, such as pyrogenic reactions, damages caused by bacteria\'s considered pathogenic in patients immunologically compromised, damages to the physical integrity of the materials, as well as prolonged permanence of the patient in the hospital, has raised interests in the evaluation of physical and biological aspects of the reused medical device. Based on these considerations, challenges had been applied with spores of Bacillus Subtilis varo niger 9372 ATCC and bacterial endotoxin E. coli 055:B5. The selected materials includes, intravenous catheters, three-way stopcocks and traqueostomy tubes. The possible presence of bacteria was investigated after intentional contamination with spores of B. Subtillis (107 cfu/unit) followed by submission of the inoculated units to the cleaning process and posterior sterilization using ethylene oxide at 12:88. The simulated reprocessing cycles of the medical device had consisted of contamination of each test unit with predetermined microbial load, washing with enzymatic detergent, drying and sterilization. At the end of each reprocessing cycle, representative sarnples had been segregated to evaluate microbial count (pour plate), terility test through direct and indirect inoculation, cytotoxicity test through cell culture and scanning electron microscopy. The efficiency of the sterility was evaluated through microbial count, as for the sterility tests, that had resulted in microbial count of 103 cfu/unit and detection of contamination until the 6th cycle of reprocessing in the intravenous catheters, three-way stopcocks and traqueostomy tubes. The safety of these reprocessed medical devices was evaluated by mouse fibroblasts cell culture (NCTC clone 929), which didn\'t presented significant toxicity. However, the results obtained by scanning electron microscopy proved the presence of microbial load after 10th reprocessing cycle, as well as damages to the polymeric surface. During challenge with bacterial endotoxin, that consisted of contaminating the units with 200 EU, drying and submission to sterilization cycles with ethylene oxide/cfe (12:88), it was verified that after ten simulated reprocessing cycles, it was possible to recover around 100% of endotoxin. The catheter guide that were acquired from institutional hospital after four times reuse, had presented levels of contamination of 105 cfu/unit, as well as presenee of bacteria considered pathogenic in patients immunologically compromised, on the other hand the detention of bacterial endotoxin in these catheters was not considered significant. The evaluations studies condueted with the units submitted to the simulated reprocessing cycles, as well as the catheter guide reprocessed and reused four times, had reflected the reality of some national and foreign hospitais which reutilize single use medical device. The results obtained lead us to raise objections to the reusing practice, considering the absence of adequate patients safety.

Neste trabalho obtemos taxas de decaimento para autovalores e valores singulares de operadores integrais gerados por núcleos de quadrado integrável sobre a esfera unitária em \'R POT. m+1\', m 2, sob hipóteses sobre ambos, certas derivadas do núcleo e o operador integral gerado por tais derivadas. Este tipo de problema é comum na literatura, mas as hipóteses geralmente são definidas via diferenciação usual em \'R POT m+1\'. Aqui, as hipóteses são todas definidas via derivada de Laplace-Beltrami, um conceito genuinamente esférico investigado primeiramente por W. Rudin no começo dos anos 50. As taxas de decaimento apresentadas são ótimas e dependem da dimensão m e da ordem de diferenciabilidade usada para definir as condições de suavidade / In this work we obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in \'R m+1\', m 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined via standard differentiation in \'R POT. m+1\'. Here, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by W. Rudin in the early fifties and genuinely spherical in nature. The rates we present are optimal and depend on both, the differentiability order used to define the smoothness conditions and the dimension m

Neste trabalho avaliamos a formação de oxigênio singlete in vitro em fagócitos, (células mononucleares e neutrófilos) isolados de sangue periférico humano, e eosinófilos, de lavado bronco alveolar de camundongos balb/c, ativados por estímulo partículado: zimosan opsonizado contendo o 9,10difenilantraceno (DPA) adsorvido como sonda captadora de 1O2. Por este método, a formação do 1O2 pode ser verificada pela formação do 9,10-difenilantraceno endoperóxido (DPAO2), que é detectado por HPLC. Observamos, que os fagócitos formam 1O2 e que esta formação parece ocorrer de forma diferenciada para os dois tipos celulares (neutrófilos e células mononucleares). Visando ampliar os estudos anteriores sobre o papel da melatonina (MLT) no processo inflamatório, foi testado seu efeito em fagócitos e a relação na produção de 1O2 destas células. Observamos que MLT inibe a formação de 1O2 totalmente no caso de neutrófilos e parcialmente no caso de células mononucleares e eosinófilos. Paralelamente, foi desenvolvida a síntese de um novo captador químico de 1O2, o éster 9,10-antracenil-3-bispropionato de etila (ABPE), cuja finalidade principal é o acúmulo no interior da célula, depois de sofrer hidrólise enzimática. Esta sonda, terá facil acesso ao interior das células em sua forma ester. Este novo captador de 1O2 foi testado em células mononucleares e neutrófilos estimulados de formas diferentes: via receptor independente e dependente. Os resultados demonstraram produção equivalente de 1O2 nestes fagócitos. / In this study, we evaluated the singlet oxygen (1O2) formation in vitro from phagocytes (neutrophils and mononuclear cells) isolated from human blood cells and eosinophils isolated from bronchoalveolar lavage fluid of mice balb/c activated, by opsonized zymosan. To determine whether singlet oxygen is produced by phagocytes, zymosan particles were coated with a specific chemical trap for 1O2, 9,10-diphenylanthracene (DPA). The production of 1O2 was followed using HPLC, to measure its product, 9,10-diphenylanthracene endoperoxide (DPAO2). We also noticed that the 1O2 production occurs at different levels of for two cell types, neutrophils and mononuclear cells. In order to broaden previous studies on the role of melatonin (MLT) in inflammatory processes, its effect was tested in phagocytes was tested in relation to 1O2 formation by these cells. We observed that MLT inhibits the 1O2 formation totallymt neutrophils and partiallym mononuclear cells and eosinophils. At the some time, it was also developed the synthesis of a new probe for 1O2, the 9,10-anthracene-bis-3-ethyl-propionate (ABEP), with the purpose to accumulate inside the cells, after its enzymatic hydrolysis. This probe presents easy acess to the inferior of the cells in its ester form. This new probe for trapping 1O2 was tested in mononuclear cells and neutrophils stimulated in two ways: via independent and dependent receptor. The results showed equivalent production of 1O2 for both cell types.