The Riesz space structure of an Abelian W*-algebra

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M₀ be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M₀, and an elementary proof is given of the fact that a positive self-adjoint transformation in M₀ has a unique positive square root in M₀. It is then shown that when the algebraic operations are suitably defined, then M₀ becomes a commutative algebra. If ReM₀ denotes the set of all self-adjoint elements of M₀, then it is proved that ReM₀ is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM₀ which characterizes the normal integrals on the order dense ideals of ReM₀. It is then shown that ReM₀ may be identified with the extended order dual of ReM, and that ReM₀ is perfect in the extended sense.

Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

A organização do trabalho na Unidade de Doenças Infecto-contagiosas e a ocorrência de Burnout nos trabalhadores de Enfermagem

Na organização do trabalho hospitalar há vários determinantes que acarretam no desgaste psicofísico do trabalhador de enfermagem, mesmo com o discurso de que gostam da profissão e se sentem realizados em cuidar de pessoas enfermas, especialmente, no cuidado de clientes adoecidos com o HIV/Aids. A Psicodinâmica do Trabalho é uma ciência que possibilita analisar a configuração da organização laboral, a qual comprovadamente incide na dimensão subjetiva do trabalhador, identificando o sofrimento psíquico, o que potencializa o desenvolvimento de doenças mentais, entre elas a Síndrome de Burnout. Nesta perspectiva, o objeto deste estudo trata da organização do trabalho na Unidade de Doença Infecto-Contagiosa, espaço de cuidado de clientes com HIV/Aids e a ocorrência de Burnout entre os trabalhadores de enfermagem que atuam neste espaço laboral. A fim de apreender o objeto traçaram-se três objetivos: a) identificar a percepção dos trabalhadores acerca das características do trabalho de enfermagem no contexto da Unidade de Doença Infecto-Contagiosa, local de assistência ao cliente portador do HIV/Aids; b) descrever as repercussões no processo saúde-doença dos trabalhadores de enfermagem decorrente da assistência ao cliente com HIV/AIDS; e c) analisar as repercussões do processo saúde-doença dos trabalhadores de enfermagem com vistas à identificação de situações do aparecimento da Síndrome de Burnout. Para a realização desta pesquisa, optou-se pela abordagem qualitativa, de caráter descritivo e exploratório. Os dados foram obtidos nos meses de maio a agosto de 2010, utilizando as seguintes fontes de coleta de informações: a entrevista semi-estruturada e o formulário Maslach Burnout Inventory. Optou-se por analisar as informações através do Método de Análise Temática de Conteúdo. Os resultados indicaram que o perfil do profissional de enfermagem era composto por trabalhadores do sexo feminino, que estavam na faixa etária entre 44 e 54 anos de idade, na grande maioria técnicos de enfermagem com tempo médio de 2 a 10 anos de trabalho com clientes HIV/Aids. Verificou-se também que havia discrepâncias marcantes entre o trabalho prescrito e o real, o que acarretava sofrimento para o profissional de enfermagem. Constatou-se também que o sofrimento psíquico resultava da vivência cotidiana do processo de morte/morrer do cliente com HIV/Aids, pelo profissional de enfermagem. Além disso, este sofrimento era determinado também pela precarização das relações e das condições de trabalho. Concluiu-se que havia vários trabalhadores com fortes indícios de ocorrência de Burnout, tanto porque a organização do trabalho se configurava como incoerente e pouco racional como pelas características do processo de cuidar do cliente com HIV/Aids. Recomendam-se medidas que promovam a saúde dos trabalhadores de enfermagem e previnam os agravos em seus processos saúde-doença, tais como: a diminuição da carga emocional de trabalho, grupos de reflexão, ginástica laboral, entre outras. É preciso haver conscientização dos gestores, vontade política e estímulo da organização laboral para que os trabalhadores participem.

On the Cesari fixed point method in a Banach space

In a paper published in 1961, L. Cesari [1] introduces a method which extends certain earlier existence theorems of Cesari and Hale ([2] to [6]) for perturbation problems to strictly nonlinear problems. Various authors ([1], [7] to [15]) have now applied this method to nonlinear ordinary and partial differential equations. The basic idea of the method is to use the contraction principle to reduce an infinite-dimensional fixed point problem to a finite-dimensional problem which may be attacked using the methods of fixed point indexes.

The following is my formulation of the Cesari fixed point method:

Let B be a Banach space and let S be a finite-dimensional linear subspace of B. Let P be a projection of B onto S and suppose Г≤B such that pГ is compact and such that for every x in PГ, P^-1x∩Г is closed. Let W be a continuous mapping from Г into B. The Cesari method gives sufficient conditions for the existence of a fixed point of W in Г.

Let I denote the identity mapping in B. Clearly y = Wy for some y in Г if and only if both of the following conditions hold:

(i) Py = PWy.

(ii) y = (P + (I - P)W)y.

Definition. The Cesari fixed paint method applies to (Г, W, P) if and only if the following three conditions are satisfied:

(1) For each x in PГ, P + (I - P)W is a contraction from P^-1x∩Г into itself. Let y(x) be that element (uniqueness follows from the contraction principle) of P^-1x∩Г which satisfies the equation y(x) = Py(x) + (I-P)Wy(x).

(2) The function y just defined is continuous from PГ into B.

(3) There are no fixed points of PWy on the boundary of PГ, so that the (finite- dimensional) fixed point index i(PWy, int PГ) is defined.

Definition. If the Cesari fixed point method applies to (Г, W, P) then define i(Г, W, P) to be the index i(PWy, int PГ).

The three theorems of this thesis can now be easily stated.

Theorem 1 (Cesari). If i(Г, W, P) is defined and i(Г, W, P) ≠0, then there is a fixed point of W in Г.

Theorem 2. Let the Cesari fixed point method apply to both (Г, W, P₁) and (Г, W, P₂). Assume that P₂P₁=P₁P₂=P₁ and assume that either of the following two conditions holds:

(1) For every b in B and every z in the range of P₂, we have that ‖b=P₂b‖ ≤ ‖b-z‖

(2)P₂Г is convex.

Then i(Г, W, P₁) = i(Г, W, P₂).

Theorem 3. If Ω is a bounded open set and W is a compact operator defined on Ω so that the (infinite-dimensional) Leray-Schauder index i_LS(W, Ω) is defined, and if the Cesari fixed point method applies to (Ω, W, P), then i(Ω, W, P) = i_LS(W, Ω).

Theorems 2 and 3 are proved using mainly a homotopy theorem and a reduction theorem for the finite-dimensional and the Leray-Schauder indexes. These and other properties of indexes will be listed before the theorem in which they are used.

Tastes and odours in potable water:perception versus reality

Tastes and odours are amongst the few water quality standards immediately apparent to a consumer and, as a result, account for most consumer complaints about water quality. Although taste and odour problems can arise from a great many sources, from an operational point of view they are either ”predictable” or ”unpredictable”. The former - which include problems related to actinomycete and algal growth - have a tendency to occur in certain types of water under certain combinations of conditions, whereas the latter - typically chemical spills - can occur anywhere. Long-term control is one option for predictable problems, although biomanipulation on a large scale has had utile success. Detection and avoidance is a more practicable option for both predictable and unpredictable problems, particularly if the distribution network can be serviced from other sources. Where these are not feasible, then water treatment, typically using activated carbon, is possible. In general there is a reasonable understanding of what compounds cause taste and odour problems, and how to treat these. An efficient taste and odour control programme therefore relies ultimately on good management of existing resources. However, a number of problems lie outside the remit of water supply companies and will require more fundamental regulation of activities in the catchment.

Temporal-space-transforming pulse-shaping system with a knife-edge apparatus for a high-energy laser facility

For the first time to our knowledge, in a high-energy laser facility with an output energy of 454.37 J, by using a temporal-space-transforming pulse-shaping system with our own design of a knife-edge apparatus, we obtained a quasi-square laser pulse. (c) 2005 Optical Society of America.