Asymptotics in a neighborhood of infinity of solutions with finite Dirichlet integral of second-order elliptic equations

In this paper we answer a question posed by E. Sanchez-Palencia which arose in the theory of homogenization of differential operators. The asymptotic behavior of solutions at infinity which have finite Dirichlet integral is studied and uniqueness theorems are also proved for exterior boundary problems for second-order elliptic equations in divergent form.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 149–163, 1987.     

From simple structure to sparse components: a review

The article begins with a review of the main approaches for interpretation the results from principal component analysis (PCA) during the last 50–60 years. The simple structure approach is compared to the modern approach of sparse PCA where interpretable solutions are directly obtained. It is shown that their goals are identical but they differ by the way they are realized. Next, the most popular and influential methods for sparse PCA are briefly reviewed. In the remaining part of the paper, a new approach to define sparse PCA is introduced. Several alternative definitions are considered and illustrated on a well-known data set. Finally, it is demonstrated, how one of these possible versions of sparse PCA can be used as a sparse alternative to the classical rotation methods.     

The determination of uncertainty levels in robust clustering of subjects with longitudinal observations using the Dirichlet process mixture

In this paper we introduce a new method to the cluster analysis of longitudinal data focusing on the determination of uncertainty levels for cluster memberships. The method uses the Dirichlet-t distribution which notably utilizes the robustness feature of the student-t distribution in the framework of a Bayesian semi-parametric approach together with robust clustering of subjects evaluates the uncertainty level of subjects memberships to their clusters. We let the number of clusters and the uncertainty levels be unknown while fitting Dirichlet process mixture models. Two simulation studies are conducted to demonstrate the proposed methodology. The method is applied to cluster a real data set taken from gene expression studies.     

Time development of a Markov process in a finite population: Application to diffusion of information

A systematic procedure of truncating the hierarchy of moment equations describing the stochastic evolution of a Markov process in a finite population is developed. The procedure makes use of the asymptotic expression for a certain higher‐order moment of the relevant probability distribution and yields finite‐size corrections to all lower‐order moments. The usefulness of the method is illustrated by applying it to study the mean and the variance of the stochastic variable n(t), the number of active spreaders at time t, in Bartholomew's model of diffusion of information. The results thus obtained are compared with the ones following from the exact probability distribution for the model (wherever known) and the agreement between the two sets of results is found to be remarkably good.     

Erratum to: Special primitive pairs in finite groups

During the review of the article “Special primitive pairs in finite groups”, an error was found in the proof of one of the preliminary lemmas. This note corrects this error and the author wishes to thank George Glauberman for pointing it out.     

Associations and dissociations with time-dependent reaction coefficients in finite polymer mixtures: The model and analytical-numerical method for solution by successive approximations

The work deals with the association and dissociation reactions with time-dependent coefficients in finite mixtures of polymers dispersed in fluid media with solid components. The polymers are regarded to be formed by identical units, polymer-forming units (PFUs) and, thus, present homopolymers. The model takes into account the porosity of the dispersion-medium/polymer-mixture system. The work derives the model for the reactions in the finite mixtures. The model presents a non-autonomous quadratic finite ODE system in a time-independent hyperplane and is based on the conservation law for the total number of PFUs. A variety of engineering applications of the derived finite-mixture model are discussed. The simplest case of the finite mixtures, i.e., the monomer-dimer mixtures with time-independent reaction coefficients is completely analyzed. An analytical-numerical (AN) method of the successive-approximations (SA) type is proposed for solving the derived model. The AN/SA method includes explicit analytical expressions for each of the approximations in terms of the preceding approximation. The method is exact in the dissociation-only case. The approximations are expected to converge if the association-reaction coefficients are not too large and the zeroth approximations are not very far from the solution. The AN/SA method comprises two sequences of the approximations. If the first one converges uniformly in the entire time axis, then the limit function is a steady-state (or “dynamic equilibrium”) solution of the non-autonomous quadratic ODE system. The second sequence presumes that the first sequence is convergent in the above mentioned sense. The second sequence is intended for calculation of the solutions of initial-value problems for the above ODE system in a semi-infinite time interval. The main differences from common computational methods are formulated. The AN/SA method is quantitatively illustrated with a few examples of the settings in the aforementioned case of monomer-dimer mixtures, also in comparison with the explicit Euler method. The form of the AN/SA method allows especially efficient implementation on multi-processor/multi-core personal computers with graphic processing units even if the dimension of the state space is large. The developed model and method form a constructive framework for analysis or design of polymer mixtures dispersed in fluid-solid media. An application to prospective manufacturing of spatially heterogeneous polymer products is noted. A few directions for future research are proposed as well.     

Optical solitons in communications: From integrability to controllability

A brief historical review and recent developments of theories and experiments of optical solitons in fibers are presented with particular emphasis on their integrability and controllability in real systems.     

Europe in symplex diagrams: From statistical tables to insight

Two-dimensional “symplex” diagrams will be presented to provide deeper insight into quantitative similarities and differences between the member states of the European Union. This paper serves two purposes: (i) It shall help to provide a better perception of quantitative European relations. (ii) It shall demonstrate the cognitive power of two-dimensional “symplex” diagrams. A collection of 12 diagrams will be used as samples.     

From centrality to temporary fame: Dynamic centrality in complex networks

We develop a new approach to the study of the dynamics of link utilization in complex networks using records of communication in a large social network. Counter to the perspective that nodes have particular roles, we find roles change dramatically from day to day. “Local hubs” have a power law degree distribution over time, with no characteristic degree value. Our results imply a significant reinterpretation of the concept of node centrality in complex networks, and among other conclusions suggest that interventions targeting hubs will have significantly less effect than previously thought. © 2006 Wiley Periodicals, Inc. Complexity 12: 59–63, 2006     

From Classroom to Online Teaching: Experiences in Improving Statistics Education

This study used reflective practitioner methodology to investigate how to improve the quality of statistical education. During the study, this methodology, curricula, pedagogical practices, assessment and a framework for learning to learn statistics were all developed as means of improving the quality of statistical education. Also documented was the move from being a classroom teacher of statistics to a teacher who is developing learning resources for online delivery to students. For a classroom teacher, flexible delivery has meant drawing on the sights, sounds, movement, quiet and live shows. By contrast, the online teacher feels the constraints of translating activity based programs to technologically based programs. As more students have chosen to rely on online materials rather than classroom activities, the focus of improving quality has been extended to the enrichment of online resources, so that the learning experience is not second to that of the classroom.     

Handling uncertainty in route choice models: From probabilistic to possibilistic approaches

In order to design effective advanced traffic information systems (ATIS) suitable mathematical models have to be defined to simulate the effects of information on users route choice behaviour and then to incorporate it into traffic assignment models to estimate how traffic demand loads the roads network.To face this problem it is necessary to deal with uncertainty that plays a crucial role in the users decision-making processes.To this purpose this paper first analyses how uncertainty affects users’ route choice process and how traffic assignment models may take it into account.In literature route choice behaviour modelling is widely solved within the random utility theory framework but, we show in this paper that such an approach only considers one type of uncertainty. More precisely, the consideration of randomness of traffic by drivers is, for example, hardly ever represented in classical models in spite of its importance in the management of information by drivers.Starting from the presented analysis a new route choice model is also proposed to represent explicitly the uncertainty lying in users’ route choice behaviour. It is based on recent developments in possibility theory which is an alternate way to probability theory in order to represent or measure uncertainty.     

From Plato to Frege: Paradigms of Predication in the History of Ideas

One of the perennial questions of philosophy concerns the simple statements which say that an object is so and so or that such and such objects are so and so related: simple predicative statements. Do such statements have an ontological basis, and if so, what is that basis? The answer to this question determines—or in any case, is expressive of—a specific fundamental outlook on the world. In the course of the history of Western philosophy, various philosophers have given various answers to the question of predication. This essay presents the main, crucial answers: the paradigms and theories of predication of the Sophists (and of all later radical relativists), of Plato, of Aristotle, of the Aristotelian-minded non-nominalists, of Leibniz, and of Frege. In addition, the essay follows (to some extent) the most influential—the Aristotelian or mereological—paradigm of predication in its continuity and modification through the many centuries of its reign. However, the essay is not content to adopt the merely historical point of view; it also poses the question of adequacy. Prior to Frege, there was no philosophically adequate theory of predication, and the essay points out the shortcomings (besides aspects that can be viewed as advantages) of each pre-Fregean predication theory considered in it. Frege, in the nineteenth century, brought the philosophy of predication on the right track, but his own theory of predication has its own deficits. The essay ends with the presentation of a theory of predication that the author himself considers adequate.