From its early beginnings onwards mankind has put to test many different society forms, and this fact raises a complex of interesting questions. The objective of this paper is to present a general population model which takes essential features of any society into account and which gives interesting answers on the basis of only two natural hypotheses. One is that societies want to survive, the second one that individuals in a society would in general like to increase their standard of living. We start by presenting a mathematical model which may be seen as a particular type of a controlled branching process. All conditions of the model are justified and interpreted.After several preliminary results about societies in general we can show that two society forms should attract particular attention, both from a qualitative and a quantitative point of view. These are the so-called weakest-first society and the strongest-first society. In particular we prove then that these two societies stand out since they form an envelope of all possible societies in a sense we will make precise. This result (the Envelopment Theorem) is seen as significant because it is paralleled with precise survival criteria for the enveloping societies. Moreover, given that one of the "limiting" societies can be seen as an extreme form of communism, and the other one as being close to an extreme version of capitalism, we conclude that, remarkably, humanity is close to having already tested the limits.
The model presented in this paper is neither a BP with varying environment (see e.g. @cite_16 ) nor a BP with random environment. See @cite_17 for a clear analysis of the connection between these two types, and e.g. @cite_24 for newer developments. Our model is no multi-type BP model, and neither a pure population size-dependent model. It is a Markov process, as we shall see, but no phase-type Markov model or decomposable BP (see @cite_20 ) can play the control we have in mind.
@cite_24 @cite_16 @cite_20 @cite_17
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We present an application of Hidden Markov Models to supervised document classification and ranking. We consider a family of models that take into account the fact that relevant documents may contain irrelevant passages; the originality of the model is that it does not explicitly segment documents but rather considers all possible segmentations in its final score. This model generalizes the multinomial Naive Bayes and it is derived from a more general model for different access tasks. The model is evaluated on the REUTERS test collection and compared to the multinomial Naive Bayes model. It is shown to be more robust with respect to the training set size and to improve the performance both for ranking and classification, specially for classes with few training examples. SUMMARY Models for the analysis of longitudinal data must recogrlize the relationship between serial observations on the same unit. Multivariate models with general covariance structure are often difficult to apply to highly unbalanced data, whereas two-stage random-effects models can be used easily. In two-stage models, the probability distributions for the response vectors of different individuals belong to a single family, but some random-effects parameters vary across individuals, with a distribution specified at the second stage. A general family of models is discussed, which includes both growth models and repeated-measures models as special cases. A unified approach to fitting these models, based on a combination of empirical Bayes and maximum likelihood estimation of model parameters and using the EM algorithm, is discussed. Two examples are taken from a current epidemiological study of the health effects of air pollution. Many longitudinal studies are designed to investigate changes over time in a characteristic which is measured repeatedly for each study participant. In medical studies, the measurement might be blood pressure, cholesterol level, lung volume, or serum glucose. Multiple measurements are obtained from each individual, at different times and possibly under changing experimental conditions. Often, we cannot fully control the circumstances under which the measurements are taken, and there may be considerable variation among individuals in the number and timing of observations. The resulting unbalanced data sets are typically not amenable to analysis using a general multivariate model with unrestricted covariance structure. Statisticians have often analyzed data of this form using some variant of a two-stage model. In this formulation, the probability distribution for the multiple measurements has the same form for each individual, but the parameters of that distribution vary over individuals. The distribution of these parameters, or 'random effects', in the population constitutes the second stage of the model. In a study of changes in lung volume during childhood, for instance, it may be reasonable to assume that the relationship between lung volume and the cube of height is linear for each child, but with linear regression parameters that vary among children. If we assume that the usual linear regression model applies for each child, conditional on the child's individual parameters, and that the regression parameters have a bivariate normal distribution in the population, the marginal distribution of the serial measurements is multivariate normal with a special covariance structure. The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007 A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for v vc. At v=vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (vc−v)−1 2 in the quasi-stationary regime below vc. Moreover for v>vc, our data indicate that there is no quasi-stationary regime.