An interdisciplinary approach to coalition formation

A stable government is by definition not dominated by any other government. However, it may happen that all governments are dominated. In graph–theoretic terms this means that the dominance graph does not possess a source. In this paper we are able to deal with this case by a clever combination of notions from different fields, such as relational algebra, graph theory and social choice theory, and by using the computer support system RelView for computing solutions and visualizing the results. Using relational algorithms, in such a case we break all cycles in each initial strongly connected component by removing the vertices in an appropriate minimum feedback vertex set. In this way we can choose a government that is as close as possible to being un-dominated. To achieve unique solutions, we additionally apply the majority ranking recently introduced by Balinski and Laraki. The main parts of our procedure can be executed using the RelView tool. Its sophisticated implementation of relations allows to deal with graph sizes that are sufficient for practical applications of coalition formation.     

Exploring informal mathematical products of low achievers at the secondary school level

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students’ products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.     

Engineering applications for demonstrating mathematical problem-solving methods at the secondary education level

A method for teaching problem-solving skills to mathematicsstudents is presented. By subtly incorporating engineering applications,the students are simultaneously exposed to an effective outreachprogramme. The process includes offering a complex problem thatchallenges students’ abilities. Before allowing the studentsto become frustrated, the students are directed to a familiarproblem, which involves an intuitive mathematical solution.The process is completed by demonstrating how the complex problemis similar to the familiar problem and solvable by the samemathematical model.     

The teaching of proof at the lower secondary level—a video study

Teaching mathematical proof is one of the most challenging topics for teachers. Several empirical studies revealed repeatedly different kinds of students’ problems in this area. The results give support that students’ abilities in proving are significantly influenced by their specific mathematics classrooms. In this paper we will present a method for evaluating proof instruction and some results of a video study that describe proving processes in mathematics classrooms at the lower secondary level from a mathematical perspective.     

Preemptive benchmarking problem: An approach for official statistics in small areas

National Statistical Agencies and Autonomous Institutions are extremely interested in using information from those areas that are actually smaller than the actual areas for which a survey is initially designed. As such, small area estimation and its application are valuable when conducting research on Official Statistics. A wide range of different methods are available which provide estimations to small area levels, being reasonable to guarantee that they add up to the published design-based estimations in a large area that includes these small areas. This requirement is known as benchmarking. Different algorithms, all based on distances between original data and modified data, are introduced in this paper, with the intention of satisfying the benchmarking property. We provide rules to apply these proposed calibrated methods according to user criteria. Goal programming with priorities methodology is used to represent user preferences. The result is a collection of different interdependent network flow problems. Some of these problems require the development of ad hoc methods. The introduced methods are assessed by a Monte Carlo simulation study using the Spanish Labour Force Survey in the Canary Islands. The results also show that the consistency of the estimator is independent of the used calibrated methods, but it does depend on the benchmarking weights.     

A copula entropy approach to correlation measurement at the country level

The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.     

A new modelling approach for the description of tissues at cell level

The understanding of tissues from a biomechanical perspective requires a deeper knowledge about cell mechanics. A lot of experimental and theoretical investigation has already been done to consider cells' behaviour under various conditions. Several types of cells exist, each with specific properties. In this work, the cell microstructure is characterized and explained via tensegrity systems. In doing so, we consider in a first step a simple cubic shaped cell built up by trusses and ropes that are discretized by finite 1D elements. This simplified microstructure represents the micro/cell level on the integration point of the finite element discretization at macro level. By using an appropriate homogenization technique for the micro level, it is assumed to gain a more detailed view on deformations occur on cell level. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An approach to hierarchical clustering via level surfaces and convexity

We consider the following approach to hierarchical data clustering. Let A = {p ₁ , p ₂ , \ldots , p _n } be a set of n data points in R ^d , d \geq 2 . Define , where z is in R^d, and |⋅ |_q denotes the L_q-norm, q \geq 1. The function f can be viewed as a ``combined luminosity' formed by summing individual ``point luminosities' located at the p_i. The level surfaces of f define a hierarchical clustering of A in a natural way. We prove a general result on convexity that enables us to obtain this clustering for q = 1 by examining the values of f on the edges of a rectilinear grid induced by A. An algorithm is developed that is practical in several situations. For n < 1000, these include: (1) d small (say d ≤ 4) and the p_i real; (2) d moderate (say d ≤ 25) and the p_i binary. Received March 10, 1999, and in revised form June 21, 2000. Online publication January 17, 2001.     

An approach to understanding avalanche statistics in mean‐field driven threshold systems

Driven threshold models that produce complex histories of avalanches are used to simulate the dynamics of many complex interacting systems, such as earthquake generating faults and neural networks. A mean‐field model may be formulated in a way that makes avalanches Abelian, so the final size of the avalanche depends only on the initial conditions, not the algorithm. If the initial stress distribution is statistically stationary, the avalanche size distribution is generated by the first intersection of a random process with a curvilinear boundary. Solutions show that such mean‐field models are never truly critical, but always exhibit dissipation or finite‐size effects. © 2005 Wiley Periodicals, Inc. Complexity 10:68–72, 2005     

Incentivizing advanced mathematics study at upper secondary level: the case of bonus points in Ireland

Secondary level mathematics education in Ireland has recently experienced a period of significant change with the introduction of new curricula and the addition of an incentive to study upper secondary mathematics at the most advanced level (Higher Level). This incentive, typically referred to as ‘bonus points’, appears to have aided a significant increase in the number of students studying upper secondary mathematics at Higher Level. However, thematic analysis of interviews with experienced upper secondary mathematics examiners and exploration of mathematics diagnostic test data outlined in this paper suggest that the difficulty of the Higher Level upper secondary mathematics final examination in Ireland has reduced since the introduction of the bonus points initiative. The sharp increase in students attempting this examination coupled with a policy of maintaining a consistent proportion of students achieving passing grades was identified as a key reason for this possible reduction in standards.     

An isogeometric boundary element approach for topology optimization using the level set method

Among the various types of structural optimization, topology has been occupying a prominent place over the last decades. It is considered the most versatile because it allows structural geometry to be determined taking into account only loading and fixing constraints. This technique is extremely useful in the design phase, which requires increasingly complex computational modeling. Modern geometric modeling techniques are increasingly focused on the use of NURBS basis functions. Consequently, it seems natural that topology optimization techniques also use this basis in order to improve computational performance. In this paper, we propose a way to integrate the isogeometric boundary techniques to topology optimization through the level set function. The proposed coupling occurs by describing the normal velocity field from the level set equation as a function of the normal shape sensitivity. This process is not well behaved in general, so some regularization technique needs to be specified. Limiting to plane linear elasticity cases, the numerical investigations proposed in this study indicate that this type of coupling allows to obtain results congruent with the current literature. Moreover, the additional computational costs are small compared to classical techniques, which makes their advantage for optimization purposes evident, particularly for boundary element method practitioners.     

Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level

ZDM – Mathematics Education - This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working...     

A new approach to the level function

This article devotes to design a hybrid Level Set Method which compromises the advantages of the level functions designed in [2] and [3].     

Algebra at the tertiary level

Inevitably, in such discussion groups, the time is too short to handle topics to the depth that they deserve. It is hoped that these notes will provide some hint of the discussions in the tertiary algebra subgroup and serve as a basis for further delibertions among participants and others.