Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov’s mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.

A mean value for pairs of integrals

Se prueba que para dos funciones continuas reales, una determinada ecuación posee solució única. Se presenta también una generalización a integrales con peso.

Functional equations and the Cauchy mean value theorem

The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy mean value theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.

Mean-value identities as an opportunity for Monte Carlo error reduction

In the Monte Carlo simulation of both lattice field theories and of models of statistical mechanics, identities verified by exact mean values, such as Schwinger-Dyson equations, Guerra relations, Callen identities, etc., provide well-known and sensitive tests of thermalization bias as well as checks of pseudo-random-number generators. We point out that they can be further exploited as control variates to reduce statistical errors. The strategy is general, very simple, and almost costless in CPU time. The method is demonstrated in the twodimensional Ising model at criticality, where the CPU gain factor lies between 2 and 4.

A Quantile-Based Probabilistic Mean Value Theorem

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.

Some Mean-Value Theorems of the Cauchy Type

2000 Mathematics Subject Classification: Primary 26A24, 26D15; Secondary 41A05

On the inverse windowed Fourier transform

The inversion problem concerning the windowed Fourier transform is considered. It is shown that, out of the infinite solutions that the problem admits, the windowed Fourier transform is the "optimal" solution according to a maximum-entropy selection criterion.

Solution-free sets for sums of binary forms

In this paper, we obtain quantitative estimates for the asymptotic density of subsets of the integer lattice Z2 that contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov’s mean value theorem applicable to binary forms.

Characterization of aqueous silver nitrate solutions for leakage tests

OBJECTIVE: To determine the pH over a period of 168 h and the ionic silver content in various concentrations and post-preparation times of aqueous silver nitrate solutions. Also, the possible effects of these factors on microleakage test in adhesive/resin restorations in primary and permanent teeth were evaluated. MATERIAL AND METHODS: A digital pHmeter was used for measuring the pH of the solutions prepared with three types of water (purified, deionized or distilled) and three brands of silver nitrate salt (Merck, Synth or Cennabras) at 0, 1, 2, 24, 48, 72, 96 and 168 h after preparation, and storage in transparent or dark bottles. Ionic silver was assayed according to the post-preparation times (2, 24, 48, 72, 96, 168 h) and concentrations (1, 5, 25, 50%) of solutions by atomic emission spectrometry. For each sample of each condition, three readings were obtained for calculating the mean value. Class V cavities were prepared with enamel margins on primary and permanent teeth and restored with the adhesive systems OptiBond FL or OptiBond SOLO Plus SE and the composite resin Filtek Z-250. After nail polish coverage, the permanent teeth were immersed in 25% or 50% AgNO3 solution and the primary teeth in 5% or 50% AgNO3 solutions for microleakage evaluation. ANOVA and the Tukey's test were used for data analyses (α=5%). RESULTS: The mean pH of the solutions ranged from neutral to alkaline (7.9±2.2 to 11.8±0.9). Mean ionic silver content differed depending on the concentration of the solution (4.75±0.5 to 293±15.3 ppm). In the microleakage test, significant difference was only observed for the adhesive system factor (p=0.000). CONCLUSIONS: Under the tested experimental conditions and based on the obtained results, it may be concluded that the aqueous AgNO3 solutions: have neutral/alkaline pH and service life of up to 168 h; the level of ionic silver is proportional to the concentration of the solution; even at 5% concentration, the solutions were capable of indicating loss of marginal seal in the composite restorations; the 3-step conventional adhesive system had better performance regarding microleakage in enamel on primary and permanent teeth.

Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential

We consider the quantum dynamics of a neutral atom Bose-Einstein condensate in a double-well potential, including many-body hard-sphere interactions. Using a mean-field factorization we show that the coherent oscillations due to tunneling are suppressed when the number of atoms exceeds a critical value. An exact quantum solution, in a two-mode approximation, shows that the mean-field solution is modulated by a quantum collapse and revival sequence.

Estudo da exposição profissional a formaldeído em laboratórios hospitalares de anatomia patológica

RESUMO - A exposição a formaldeído é reconhecidamente um dos mais importantes factores de risco presente nos laboratórios hospitalares de anatomia patológica. Neste contexto ocupacional, o formaldeído é utilizado em solução, designada comummente por formol. Trata-se de uma solução comercial de formaldeído, normalmente diluída a 10%, sendo pouco onerosa e, por esse motivo, a eleita para os trabalhos de rotina em anatomia patológica. A solução é utilizada como fixador e conservante do material biológico, pelo que as peças anatómicas a serem processadas são previamente impregnadas. No que concerne aos efeitos para a saúde do formaldeído, os efeitos locais parecem apresentar um papel mais importante comparativamente com os efeitos sistémicos, devido à sua reactividade e rápido metabolismo nas células da pele, tracto gastrointestinal e pulmões. Da mesma forma, a localização das lesões correspondem principalmente às zonas expostas às doses mais elevadas deste agente químico, ou seja, o desenvolvimento dos efeitos tóxicos dependerá mais da intensidade da dose externa do que da duração da exposição. O efeito do formaldeído no organismo humano mais facilmente detectável é a acção irritante, transitória e reversível sobre as mucosas dos olhos e aparelho respiratório superior (naso e orofaringe), o que acontece em geral para exposições frequentes e superiores a 1 ppm. Doses elevadas são citotóxicas e podem conduzir a degenerescência e necrose das mucosas e epitélios. No que concerne aos efeitos cancerígenos, a primeira avaliação efectuada pela International Agency for Research on Cancer data de 1981, actualizada em 1982, 1987, 1995 e 2004, considerando-o como um agente cancerígeno do grupo 2A (provavelmente carcinogénico). No entanto, a mais recente avaliação, em 2006, considera o formaldeído no Grupo 1 (agente carcinogénico) com base na evidência de que a exposição a este agente é susceptível de causar cancro nasofaríngeo em humanos. Constituiu objectivo principal deste estudo caracterizar a exposição profissional a formaldeído nos laboratórios hospitalares de anatomia patológica Portugueses. Pretendeu-se, ainda, descrever os fenómenos ambientais da contaminação ambiental por formaldeído e explorar eventuais associações entre variáveis. Considerou-se uma amostra de 10 laboratórios hospitalares de anatomia patológica, avaliada a exposição dos três grupos profissionais por comparação com os dois referenciais de exposição e, ainda, conhecidos os valores de concentração máxima em 83 actividades. Foram aplicados simultaneamente dois métodos distintos de avaliação ambiental: um dos métodos (Método 1) fez uso de um equipamento de leitura directa com o princípio de medição por Photo Ionization Detection, com uma lâmpada de 11,7 eV e, simultaneamente, realizou-se o registo da actividade. Este método disponibilizou dados para o referencial de exposição da concentração máxima; o outro método (Método 2) traduziu-se na aplicação do método NIOSH 2541, implicando o uso de bombas de amostragem eléctricas de baixo caudal e posterior processamento analítico das amostras por cromatografia gasosa. Este método, por sua vez, facultou dados para o referencial de exposição da concentração média ponderada. As estratégias de medição de cada um dos métodos e a definição dos grupos de exposição existentes neste contexto ocupacional, designadamente os Técnicos de Anatomia Patológica, os Médicos Anatomo-Patologistas e os Auxiliares, foram possíveis através da informação disponibilizada pelas técnicas de observação da actividade da análise (ergonómica) do trabalho. Estudaram-se diversas variáveis independentes, nomeadamente a temperatura ambiente e a humidade relativa, a solução de formaldeído utilizada, as condições de ventilação existentes e o número médio de peças processadas por dia em cada laboratório. Para a recolha de informação sobre estas variáveis foi preenchida, durante a permanência nos laboratórios estudados, uma Grelha de Observação e Registo. Como variáveis dependentes seleccionaram-se três indicadores de contaminação ambiental, designadamente o valor médio das concentrações superiores a 0,3 ppm em cada laboratório, a Concentração Média Ponderada obtida para cada grupo de exposição e o Índice do Tempo de Regeneração de cada laboratório. Os indicadores foram calculados e definidos através dos dados obtidos pelos dois métodos de avaliação ambiental aplicados. Baseada no delineado pela Universidade de Queensland, foi ainda aplicada uma metodologia de avaliação do risco de cancro nasofaríngeo nas 83 actividades estudadas de modo a definir níveis semi-quantitativos de estimação do risco. Para o nível de Gravidade considerou-se a informação disponível em literatura científica que define eventos biológicos adversos, relacionados com o modo de acção do agente químico e os associa com concentrações ambientais de formaldeído. Para o nível da Probabilidade utilizou-se a informação disponibilizada pela análise (ergonómica) de trabalho que permitiu conhecer a frequência de realização de cada uma das actividades estudadas. A aplicação simultânea dos dois métodos de avaliação ambiental resultou na obtenção de resultados distintos, mas não contraditórios, no que concerne à avaliação da exposição profissional a formaldeído. Para as actividades estudadas (n=83) verificou-se que cerca de 93% dos valores são superiores ao valor limite de exposição definido para a concentração máxima (VLE-CM=0,3 ppm). O “exame macroscópico” foi a actividade mais estudada e onde se verificou a maior prevalência de resultados superiores ao valor limite (92,8%). O valor médio mais elevado da concentração máxima (2,04 ppm) verificou-se no grupo de exposição dos Técnicos de Anatomia Patológica. No entanto, a maior amplitude de resultados observou-se no grupo dos Médicos Anatomo-Patologistas (0,21 ppm a 5,02 ppm). No que respeita ao referencial da Concentração Média Ponderada, todos os valores obtidos nos 10 laboratórios estudados para os três grupos de exposição foram inferiores ao valor limite de exposição definido pela Occupational Safety and Health Administration (TLV-TWA=0,75 ppm). Verificou-se associação estatisticamente significativa entre o número médio de peças processadas por laboratório e dois dos três indicadores de contaminação ambiental utilizados, designadamente o valor médio das concentrações superiores a 0,3 ppm (p=0,009) e o Índice do Tempo de Regeneração (p=0,001). Relativamente à temperatura ambiente não se observou associação estatisticamente significativa com nenhum dos indicadores de contaminação ambiental utilizados. A humidade relativa apresentou uma associação estatisticamente significativa apenas com o indicador de contaminação ambiental da Concentração Média Ponderada de dois grupos de exposição, nomeadamente com os Médicos Anatomo-Patologistas (p=0,02) e os Técnicos de Anatomia Patológica (p=0,04). A aplicação da metodologia de avaliação do risco nas 83 actividades estudadas permitiu verificar que, em cerca de dois terços (35%), o risco foi classificado como (pelo menos) elevado e, ainda, constatar que 70% dos laboratórios apresentou pelo menos 1 actividade com a classificação de risco elevado. Da aplicação dos dois métodos de avaliação ambiental e das informações obtidas para os dois referenciais de exposição pode concluir-se que o referencial mais adequado é a Concentração Máxima por estar associado ao modo de actuação do agente químico. Acresce, ainda, que um método de avaliação ambiental, como o Método 1, que permite o estudo das concentrações de formaldeído e simultaneamente a realização do registo da actividade, disponibiliza informações pertinentes para a intervenção preventiva da exposição por permitir identificar as actividades com a exposição mais elevada, bem como as variáveis que a condicionam. As peças anatómicas apresentaram-se como a principal fonte de contaminação ambiental por formaldeído neste contexto ocupacional. Aspecto de particular interesse, na medida que a actividade desenvolvida neste contexto ocupacional e, em particular na sala de entradas, é centrada no processamento das peças anatómicas. Dado não se perspectivar a curto prazo a eliminação do formaldeído, devido ao grande número de actividades que envolvem ainda a utilização da sua solução comercial (formol), pode concluir-se que a exposição a este agente neste contexto ocupacional específico é preocupante, carecendo de uma intervenção rápida com o objectivo de minimizar a exposição e prevenir os potenciais efeitos para a saúde dos trabalhadores expostos. ---------------- ABSTRACT - Exposure to formaldehyde is recognized as one of the most important risk factors present in anatomy and pathology laboratories from hospital settings. In this occupational setting, formaldehyde is used in solution, typically diluted to 10%, and is an inexpensive product. Because of that, is used in routine work in anatomy and pathology laboratories. The solution is applied as a fixative and preservative of biological material. Regarding formaldehyde health effects, local effects appear to have a more important role compared with systemic effects, due to his reactivity and rapid metabolism in skin, gastrointestinal tract and lungs cells. Likewise, lesions location correspond mainly to areas exposed to higher doses and toxic effects development depend more on external dose intensity than exposure duration. Human body formaldehyde effect more easily detectable is the irritating action, transient and reversible on eyes and upper respiratory tract (nasal and throat) membranes, which happen in general for frequent exposure to concentrations higher than 1 ppm. High doses are cytotoxic and can lead to degeneration, and also to mucous membranes and epithelia necrosis. With regard to carcinogenic effects, first assessment performed by International Agency for Research on Cancer in 1981, updated in 1982, 1987, 1995 and 2004, classified formaldehyde in Group 2A (probably carcinogenic). However, most recent evaluation in 2006, classifies formaldehyde carcinogenic (Group 1), based on evidence that exposure to this agent is likely to cause nasopharyngeal cancer in humans. This study principal objective was to characterize occupational exposure to formaldehyde in anatomy and pathology hospital laboratories, as well to describe formaldehyde environmental contamination phenomena and explore possible associations between variables. It was considered a sample of 10 hospital pathology laboratories, assessed exposure of three professional groups for comparison with two exposure metrics, and also knows ceiling concentrations in 83 activities. Were applied, simultaneously, two different environmental assessment methods: one method (Method 1) using direct reading equipment that perform measure by Photo Ionization Detection, with 11,7 eV lamps and, simultaneously, make activity description and film. This method provided data for ceiling concentrations for each activity study (TLV-C). In the other applied method (Method 2), air sampling and formaldehyde analysis were performed according to NIOSH method (2541). This method provided data average exposure concentration (TLV-TWA). Measuring and sampling strategies of each methods and exposure groups definition (Technicians, Pathologists and Assistants) was possible by information provided by activities (ergonomic) analysis. Several independent variables were studied, including temperature and relative humidity, formaldehyde solution used, ventilation conditions, and also anatomic pieces mean value processed per day in each laboratory. To register information about these variables was completed an Observation and Registration Grid. Three environmental contamination indicators were selected has dependent variables namely: mean value from concentrations exceeding 0,3 ppm in each laboratory, weighted average concentration obtained for each exposure group, as well each laboratory Time Regeneration Index. These indicators were calculated and determined through data obtained by the two environmental assessment methods. Based on Queensland University proposal, was also applied a methodology for assessing nasopharyngeal cancer risk in 83 activities studied in order to obtain risk levels (semi-quantitative estimation). For Severity level was considered available information in scientific literature that defines biological adverse events related to the chemical agent action mode, and associated with environment formaldehyde concentrations. For Probability level was used information provided by (ergonomic) work analysis that helped identifies activity frequency. Environmental assessment methods provide different results, but not contradictory, regarding formaldehyde occupational exposure evaluation. In the studied activities (n=83), about 93% of the values were above exposure limit value set for ceiling concentration in Portugal (VLE-CM = 0,3 ppm). "Macroscopic exam" was the most studied activity, and obtained the higher prevalence of results superior than 0,3 ppm (92,8%). The highest ceiling concentration mean value (2,04 ppm) was obtain in Technicians exposure group, but a result wider range was observed in Pathologists group (0,21 ppm to 5,02 ppm). Concerning Method 2, results from the three exposure groups, were all lower than limit value set by Occupational Safety and Health Administration (TLV-TWA=0,75ppm). There was a statistically significant association between anatomic pieces mean value processed by each laboratory per day, and two of the three environmental contamination indicators used, namely average concentrations exceeding 0,3 ppm (p=0,009) and Time Regeneration Index (p=0,001). Temperature was not statistically associated with any environmental contamination used indicators. Relative humidity had a statistically significant association only with one environmental contamination indicator, namely weighted average concentration, particularly with Pathologists group (p=0,02) and Technicians group (p=0,04). Risk assessment performed in the 83 studied activities showed that around two thirds (35%) were classified as (at least) high, and also noted that 70% of laboratories had at least 1 activity with high risk rating. The two environmental assessment methods application, as well information obtained from two exposure metrics, allowed to conclude that most appropriate exposure metric is ceiling concentration, because is associated with formaldehyde action mode. Moreover, an environmental method, like Method 1, which allows study formaldehyde concentrations and relates them with activity, provides relevant information for preventive information, since identifies the activity with higher exposure, as well variables that promote exposure. Anatomic pieces represent formaldehyde contamination main source in this occupational setting, and this is of particular interest because all activities are focused on anatomic pieces processing. Since there is no prospect, in short term, for formaldehyde use elimination due to large number of activities that still involve solution use, it can be concluded that exposure to this agent, in this particular occupational setting, is preoccupant, requiring an rapid intervention in order to minimize exposure and prevent potential health effects in exposed workers.