Study of the damage-induced anisotropy of quasi-brittle materials using the component assembling model

Damage-induced anisotropy of quasi-brittle materials is investigated using component assembling model in this study. Damage-induced anisotropy is one significant character of quasi-brittle materials coupled with nonlinearity and strain softening. Formulation of such complicated phenomena is a difficult problem till now. The present model is based on the component assembling concept, where constitutive equations of materials are formed by means of assembling two kinds of components' response functions. These two kinds of components, orientational and volumetric ones, are abstracted based on pair-functional potentials and the Cauchy - Born rule. Moreover, macroscopic damage of quasi-brittle materials can be reflected by stiffness changing of orientational components, which represent grouped atomic bonds along discrete directions. Simultaneously, anisotropic characters are captured by the naturally directional property of the orientational component. Initial damage surface in the axial-shear stress space is calculated and analyzed. Furthermore, the anisotropic quasi-brittle damage behaviors of concrete under uniaxial, proportional, and nonproportional combined loading are analyzed to elucidate the utility and limitations of the present damage model. The numerical results show good agreement with the experimental data and predicted results of the classical anisotropic damage models.

煤与瓦斯突出煤岩初应力波动效应研究

基于Biot多孔介质波动理论和Cauchy初应力分析结果给出横观各向同性介质初应力存在的本构方程。利用位移标量势函数得到介质中膨胀波传播的四个波速。通过数值计算研究初应力和孔隙压力存在对膨胀波传播特性的影响,以及波速随流体黏性和应力波频率的变化规律。

Poverty and aquatic resources in Vietnam: an assessment of the role and potential of aquatic resource management in poor people's livelihoods

This report seeks to discuss a variety of approaches to poverty in order to illustrate the diversity of poor people, and the range of ways in which people are poor, facilitating a broader understanding of poverty and the significance of aquatic resources in poor people’s livelihoods. This is intended to provide a balance to the general neglect of the poor in the pursuit of aquaculture development within the Fisheries sector. It is also intended that this approach to poverty will assist in the planning and targeting of aquatic resource interventions that aim to promote poverty alleviation. In its many different forms, poverty remains a persistent problem with a great number of people facing deprivation and vulnerable livelihoods. Rates of poverty alleviation also differ; whereas the Red River Delta has achieved the greatest reductions in poverty, the Mekong Delta has achieved the smallest improvements, with possible indications that inequality has increased (see 2.4). Inequality between regions persists despite progress in all regions. (PDF has 37 pages.)

土的本构模型及其工程应用

第一章 绪论
1、1土的本构特性
1、2土本构模型的发展简史
1、3土本构模型的研究动向
2、1应力分析
第二章 连续介质力学的基本概念
2、1、1一点的应力状态、应力张量
2、1、2Cauchy公式、求和协定
2、1、3主应力
2、1、4偏应力
2、1、5八面体应力、纯剪应力、主剪应力
2、1、6应力空间、应力路径
2、1、7应力Mohr圆和应力Lode参数
2、2应变分析
2、2、1一点的应变状态、应变张量
2、2、2应变Cauchy公式
2、2、3主应变
2、2、4偏应变
2、2、5八面体应变、纯应变、主剪应变
2、2、6应变空间、应变路径
2、2、7应变率张量、应变增量张量
2、2、8应变Mohr圆
2、2、9有限应变
2、3基本方程
2、3、1连续方程
2、3、2运动微分方程
2、3、3协调方程
2、3、4能量方程
2、3、5本构方程
2、3、6边界条件和初始条件
第三章 经典塑性理论简述
3、1屈服准则
3、1、1初始屈服
3、1、2后继屈服
3、1、3几种屈服条件
3、2加载和卸载准则
3、2、1理想塑性材料的加载和卸载
3、2、2硬化材料的加载和卸载准则
3、3硬化规律
3、3、1各向同性硬化模型
3、3、2随动硬化模型
3、3、3混合硬化模型
3、4塑性公设
3、4、1Drucker塑性公设
3、4、2Ильюшин塑性公设
3、5流动规则
3、5、1塑性位势理论的基本概念
3、5、2流动规则
3、6塑性形变理论与塑性增量理论
3、6、1塑性形变理论
3、6、2塑性增量理论
第四章 土的弹性本构模型
4、1线弹性模型
4、1、1广义Hook定律
4、1、2正交各向异性线弹性体
4、1、3横观各向同性线弹性体
4、1、4各向同性线弹性体
4、2应变能和应变余能
4、3能量正定性与弹性材料稳定性
4、4具有割线模量的非线性弹性模型
4、4、1全量型应力—应变关系
4、4、2增量型应力—应变关系
4、5Cauchy弹性模型
4、5、1全量型Cauchy弹性模型应力—应变关系
4、5、2增量型Cauchy弹性模型应力—应变关系
4、6超弹性模型
4、6、1全量型超弹性模型应力—应变关系
4、6、2增量型超弹性模型的应力—应变关系
4、7次弹性模型
4、8结语
5、1本构关系的普遍表达式
第五章 土的弹性—理想塑性模型
5、2本构模型中材料常数的确定
5、3本构模型的数值计算
5、4Prandtl—Reuss模型
5、5Drucker—Prager模型
5、6Coulomb模型
6、1本构关系的普遍表达式
第六章 土的弹性—硬化塑性模型
6、2剑桥模型
6、3修正剑桥模型
6、4Lade—Duncan模型
6、5帽盖模型
6、5、1一般增量应力—应变关系与刚度矩阵的推导
6、5、2模型的拟合过程
6、5、3帽盖模型的数值计算
第七章 土的粘弹塑性模型
7、1土的流变学基本模型
7、2Maxwell体模型
7、3Kelvin体模型
7、4粘塑性体模型
7、5三元模型
7、6多元件组合模型
8、1弹塑性横观各向同性模型
第八章 土本构模型的近期发展
8、2非线性弹性—硬化塑性帽盖模型
8、3弹/粘塑性动态帽盖模型
8、4多重屈服面模型
8、5边界面模型
8、6内时本构方程
9、1基础的沉降与塌陷
第九章 土本构模型在工程中的应用
9、1、1具有不同材料常数的Drucker—Prager模型
9、1、2具有非相关联流动的Drucker—Prager模型
9、1、3具有相关流动的帽盖模型
9、2堤坝的非线性分析
9、3基坑开挖的非线性分析
9、3、1基坑竣工后状况
9、3、2边坡对地震过程的响应
9、3、3地震后的滑移
参考文献

Assessing the sustainability of brackish water aquaculture systems in the Philippines. Working Paper 4: Aquaculture and poverty - a case study of five coastal communities in the Philippines

After reviewing the rather thin literature on the subject, we investigate the relationship between aquaculture and poverty based on a case study of five coastal communities in the Philippines. The analysis relies on a data set collated through a questionnaire survey of 148 households randomly selected in these five communities. The methodological approach combines the qualitative analysis of how this relationship is perceived by the surveyed households and a quantitative analysis of the levels and determinants of poverty and inequality in these communities. There is overwhelming evidence that aquaculture benefits the poor in important ways and that it is perceived very positively by the poor and non-poor alike. In particular, the poor derive a relatively larger share of their income from aquaculture than the rich, and a lowering of the poverty line only reinforces this result. Further, a Gini decomposition exercise shows unambiguously that aquaculture represents an inequality-reducing source of income. We believe that the pro-poor character of brackish water aquaculture in the study areas is explained by the fact that the sector provides employment to a large number of unskilled workers in communities characterized by large surpluses of labour. Our results also suggest that the analysis of the relationship between aquaculture and poverty should not focus exclusively on the socio-economic status of the farm operator/owner, as has often been the case in the past. [PDF contains 51 pages]

基于图像处理的陶瓷基复合材料的微结构表征

本文综述了目前边界元法研究现状和进展，分析了边界元法长期以来存在着超奇异积分和几乎超奇异积分的计算难题，该问题一直困扰着边界元法的应用范围和效率。文中针对二维边界积分方程中几乎奇异积分问题，取用二节点线性单元，剖析了边界积分方程中出现几乎奇异积分的根源。提出了接近度概念，定量地度量了单元上积分发生几乎奇异性的程度。作者寻找到一些积公式，采用分部积分法将几乎奇异积分转化为无奇异积分和解析积分之和，从而获得正则化算法。针对三维边界元法中几乎奇异面积分问题，取用三角形线性单元，在三角形平面内采用极坐标（ρ，θ），建立一种半解析算法，对变量ρ施用分部积分法将几乎强奇异和超奇异面积分转化为沿单元围道的一系列线积分，然后Gauss数值积分能够胜任这些线积的计算。对于高阶单元，提出将高阶单元细分为若干线性单元策略进行处理。对二、三维问题的几乎奇异积分分别给出了数值实验，即使接近度非常小，本文方法计算值与精确值非常一致。本文将正则化算法和半解析算法运用于二、三维弹性力学问题边界元法中，直接地计算出单元上的几乎强奇异和超奇异积分，成功地求解了近边界点的位移和应力。与已有算法比较，本文方法简单，易于施行，精度高。同样，本文方法在边界方法计算位势问题中几乎超奇异积分也获得成功。另则，因为几乎奇异积分的障碍，一种观点认为边界元法无力求解薄壁弱性体问题，本文的正则化算法同样成功地计算了源点在边界时的边界积分方程中几乎奇异积分，显示了边元法能够有效地分析薄壁结构及组合结构。导数场边界积分方程中存在着超奇异主值积分的计算屏障。对于弹性力学平面问题，本文提出以符号算子δ_(ij)和∈_(ij)（排列张量）分别作用于位移导数边界积分方程，运用一系列数学技巧将边界位移、面力和位移导数变换为新的边界变量，从而获得一个新的导数场边界积分方程－自然边界积分方程。自然边界积分方程仅存在Cauchy主值积分，文中导出了相应的主值积分列式和奇性系数。自然边界积分方程与位移边界积分方程联合可直接获取边界应力，并且精度与位移相当。自然边界积分方程的一个优点是可以仅在我们感兴趣的局部边界段建立求解。文中提出联合位移边界积分方程和自然边界积分方程计算二维弹性裂纹体的位移场、应力场和应力强度因子，结合几乎奇异积分的正则化算法，求解了含狭窄孔洞的高度应力集中问题。若干算例显示了本文方法计算结果与分析解吻合的很好。因为位移边界积分方程在裂纹面上是不适定的，本文建议采用位移边界积分方程、自然边界积分方程和差分关系联立求解一般的二维断裂力学问题的位移场和应力场，由此求得应力强的因子。

The elasto-damage theory of the components assembling model

The potential energy in materials is well approximated by pair functional which is composed of pair potentials and embedding energy. During calculating material potential energy, the orientational component and the volumetric component are derived respectively from pair potentials and embedding energy. The sum of energy of all these two kinds of components is the material potential. No matter how microstructures change, damage or fracture, at the most level, they are all the changing and breaking atomic bonds. As an abstract of atomic bonds, these components change their stiffness during damaging. Material constitutive equations have been formulated by means of assembling all components' response functions. This material model is called the component assembling model. Theoretical analysis and numerical computing indicate that the proposed model has the capacity of reproducing some results satisfactorily, with the advantages of great conceptual simplicity, physical explicitness, and intrinsic induced anisotropy, etc.

Parallel Fully-Implicit Computation of MHD Acceleration Experiments

A three-dimensional MHD solver is described in the paper. The solver simulates reacting flows with nonequilibrium between translational-rotational, vibrational and electron translational modes. The conservation equations are discretized with implicit time marching and the second-order modified Steger-Warming scheme, and the resulted linear system is solved iteratively with Newton-Krylov-Schwarz method that is implemented by PETSc package. The results of convergence tests are plotted, which show good scalability and convergence around twice faster when compared with the DPLR method. Then five test runs are conducted simulating the experiments done at the NASA Ames MHD channel, and the calculated pressures, temperatures, electrical conductivity, back EMF, load factors and flow accelerations are shown to agree with the experimental data. Our computation shows that the electrical conductivity distribution is not uniform in the powered section of the MHD channel, and that it is important to include Joule heating in order to calculate the correct conductivity and the MHD acceleration.

Nucleon structure functions from ν_µ-Fe interactions and a study of the valence quark distribution

Data were taken in 1979-80 by the CCFRR high energy neutrino experiment at Fermilab. A total of 150,000 neutrino and 23,000 antineutrino charged current events in the approximate energy range 25 < E_v < 250GeV are measured and analyzed. The structure functions F2 and xF_3 are extracted for three assumptions about σ_L/σ_T:R=0., R=0.1 and R= a QCD based expression. Systematic errors are estimated and their significance is discussed. Comparisons or the X and Q^2 behaviour or the structure functions with results from other experiments are made.

We find that statistical errors currently dominate our knowledge of the valence quark distribution, which is studied in this thesis. xF_3 from different experiments has, within errors and apart from level differences, the same dependence on x and Q^2, except for the HPWF results. The CDHS F_2 shows a clear fall-off at low-x from the CCFRR and EMC results, again apart from level differences which are calculable from cross-sections.

The result for the the GLS rule is found to be 2.83±.15±.09±.10 where the first error is statistical, the second is an overall level error and the third covers the rest of the systematic errors. QCD studies of xF_3 to leading and second order have been done. The QCD evolution of xF_3, which is independent of R and the strange sea, does not depend on the gluon distribution and fits yield

ʌ_(LO) = 88^(+163)_(-78) ^(+113)_(-70) MeV

The systematic errors are smaller than the statistical errors. Second order fits give somewhat different values of ʌ, although α_s (at Q^2_0 = 12.6 GeV^2) is not so different.

A fit using the better determined F_2 in place of xF_3 for x > 0.4 i.e., assuming q = 0 in that region, gives

ʌ_(LO) = 266^(+114)_(-104) ^(+85)_(-79) MeV

Again, the statistical errors are larger than the systematic errors. An attempt to measure R was made and the measurements are described. Utilizing the inequality q(x)≥0 we find that in the region x > .4 R is less than 0.55 at the 90% confidence level.

A geometric analysis of convex demixing

Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.

Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.

The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.

Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Fracture of materials undergoing solid-solid phase transformation

A large number of technologically important materials undergo solid-solid phase transformations. Examples range from ferroelectrics (transducers and memory devices), zirconia (Thermal Barrier Coatings) to nickel superalloys and (lithium) iron phosphate (Li-ion batteries). These transformations involve a change in the crystal structure either through diffusion of species or local rearrangement of atoms. This change of crystal structure leads to a macroscopic change of shape or volume or both and results in internal stresses during the transformation. In certain situations this stress field gives rise to cracks (tin, iron phosphate etc.) which continue to propagate as the transformation front traverses the material. In other materials the transformation modifies the stress field around cracks and effects crack growth behavior (zirconia, ferroelectrics). These observations serve as our motivation to study cracks in solids undergoing phase transformations. Understanding these effects will help in improving the mechanical reliability of the devices employing these materials.

In this thesis we present work on two problems concerning the interplay between cracks and phase transformations. First, we consider the directional growth of a set of parallel edge cracks due to a solid-solid transformation. We conclude from our analysis that phase transformations can lead to formation of parallel edge cracks when the transformation strain satisfies certain conditions and the resulting cracks grow all the way till their tips cross over the phase boundary. Moreover the cracks continue to grow as the phase boundary traverses into the interior of the body at a uniform spacing without any instabilities. There exists an optimal value for the spacing between the cracks. We ascertain these conclusion by performing numerical simulations using finite elements.

Second, we model the effect of the semiconducting nature and dopants on cracks in ferroelectric perovskite materials, particularly barium titanate. Traditional approaches to model fracture in these materials have treated them as insulators. In reality, they are wide bandgap semiconductors with oxygen vacancies and trace impurities acting as dopants. We incorporate the space charge arising due the semiconducting effect and dopant ionization in a phase field model for the ferroelectric. We derive the governing equations by invoking the dissipation inequality over a ferroelectric domain containing a crack. This approach also yields the driving force acting on the crack. Our phase field simulations of polarization domain evolution around a crack show the accumulation of electronic charge on the crack surface making it more permeable than was previously believed so, as seen in recent experiments. We also discuss the effect the space charge has on domain formation and the crack driving force.

Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Singularities and horizons in the collisions of gravitational waves

This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitational waves, as described by classical general relativity. It is focused mainly on two fundamental questions: First, what is the general structure of the singularities and Killing-Cauchy horizons produced in the collisions of exactly plane-symmetric gravitational waves? Second, under what conditions will the collisions of almost-plane gravitational waves (waves with large but finite transverse sizes) produce singularities?

In the work on the collisions of exactly-plane waves, it is shown that Killing horizons in any plane-symmetric spacetime are unstable against small plane-symmetric perturbations. It is thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactly plane gravitational waves are nongeneric, and that generic initial data for the colliding plane waves always produce "pure" spacetime singularities without such horizons. This conclusion is later proved rigorously (using the full nonlinear theory rather than perturbation theory), in connection with an analysis of the asymptotic singularity structure of a general colliding plane-wave spacetime. This analysis also proves that asymptotically the singularities created by colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes and exponents of these singularities in general depend on the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

In the work on collisions of almost-plane gravitational waves, first some general properties of single almost-plane gravitational-wave spacetimes are explored. It is shown that, by contrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagation direction that is Killing; i.e., it must diffract and disperse as it propagates. It is also shown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts; i.e., it must leave behind tails in the spacetime region through which it passes. Next, the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initially exactly plane-symmetric across a central region of sufficiently large but finite transverse dimensions, then their collision produces a spacetime singularity with the same local structure as in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when the central regions are only approximately plane-symmetric initially. Stated more precisely, it is proved that if the colliding almost-plane waves are initially sufficiently close to being exactly plane-symmetric across a bounded central region of sufficiently large transverse dimensions, then their collision necessarily produces spacetime singularities. In this case, nothing is now known about the local and global structures of the singularities.

A mulher no mercado de trabalho em Portugal e no Brasil: a procura de um espaço de igualdade de oportunidades

O presente trabalho académico aborda a situação e a relação da mulher no mercado de trabalho. Propomo-nos perceber como funciona a dinâmica do mundo do trabalho e que impacto traz nas relações sociais do indivíduo e na identidade de um país. A questão fundamental deste estudo, é se a desigualdade de oportunidades afecta diferentemente em relação ao género e como é vivida essa situação em cada um dos países estudados: Portugal e Brasil. Centramo-nos especificamente, nos seguintes objectivos: identificar os agentes que determinam a existência da discriminação sexual no mercado de trabalho, reforçar a importância de combater situações discriminatórias para o desenvolvimento de sociedades benéficas, justas e equitativas e finalmente promover o intercâmbio de conhecimentos entre Portugal e Brasil. Com este trabalho de pesquisa teórico-histórica e empírica, concluímos que a desigualdade de oportunidades existe em ambos os países participantes. Deriva primordialmente, de factores económicos, históricos e culturais ainda enraizados nas sociedades actuais. Nitidamente a mulher, ainda hoje, é vítima de uma entrada e presença no mercado de trabalho mais difícil e precária comparativamente ao homem.