The Orbit Rigidity Matrix of a Symmetric Framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.     

On the influence of theory on research in mathematics education: the case of teaching and learning limits of functions

After an introduction on approaches, research frameworks and theories in mathematics education research, three didactical research studies on limits of functions with different research frameworks are analysed and compared with respect to their theoretical perspectives. It is shown how a chosen research framework defines the world in which the research lives, pointing to the difficult but necessary task to compare research results within a common field of study but conducted within different frameworks.     

Toward networking three theoretical approaches: the case of social interactions

We rely on discussions initiated at CERME4 and continued at CERME5 to compare, contrast and combine in a coherent way different theoretical frameworks currently used in mathematics education, with the eventual aim of networking between theoretical approaches. Specifically, we chose for this purpose the theory of didactic situations (TDS), the nested epistemic actions (RBC+C) model for abstraction in context (AiC), and the theoretical approach of interest-dense situations (IDS). As an example, we focus on how each of these frameworks is taking into account social interactions in learning processes. We identified not only connections and contrasts between the frameworks but also additional insights, which each of these frameworks can provide to each of the others. We also present some methodological reflections about the process of networking different theoretical approaches.     

Boundedness, rigidity and global rigidity of direction�Clength frameworks

We describe characterizations of boundedness and rigidity of generic 2-dimensional direction?Clength frameworks, and give partial results for the open problem of characterizing the global rigidity of such frameworks.     

Rigidity and the Alexandrov-Fenchel inequality

New proofs are given for Cauchy's and Alexandrov's classical theorems on the rigidity of polyhedral frameworks, as well as their higher dimensional generalizations. Through duality, the rigidity of these frameworks follows from characterizations of the case of equality in Minkowski's quadratic inequality.     

Combining globally rigid frameworks

It is shown how to combine two generically globally rigid bar frameworks in d-space to get another generically globally rigid framework. The construction is to identify d+ 1 vertices from each of the frameworks and erase one of the edges that they have in common.     

Bipolarity in argumentation graphs: Towards a better understanding

Different abstract argumentation frameworks have been used for various applications within multi-agents systems. Among them, bipolar frameworks make use of both attack and support relations between arguments. However, there is no single interpretation of the support, and the handling of bipolarity cannot avoid a deeper analysis of the notion of support.In this paper we consider three recent proposals for specializing the support relation in abstract argumentation: the deductive support, the necessary support and the evidential support. These proposals have been developed independently within different frameworks. We restate these proposals in a common setting, which enables us to undertake a comparative study of the modellings obtained for the three variants of the support. We highlight relationships and differences between these variants, namely a kind of duality between the deductive and the necessary interpretations of the support.     

Accept & reject statement-based uncertainty models

We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss—as a bridge to existing frameworks—a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.     

A correspondence between scene analysis and motions of frameworks

For over 100 years there has been a known correspondence between plane pictures of spherical polyhedra and static stresses in bar and joint frameworks with planar graphs. We define a new map from any plane picture of a general scene to a corresponding plate and bar framework in the plane. This map, and its return map, give a simple isomorphism between the space of scenes over the picture and the space of instantaneous motions of the framework. When two simple equivalence relations on the pictures and on the frameworks are introduced, the maps define an isomorphism of the equivalence clases which permits the full translation of questions, techniques and theorems between the two fields of study. A direct geometric interpretation of this correspondence is described and an extension to systems with occlusion and tensegrity (systems of inequalities) is presented. The analogous patterns in higher dimensions are described, yielding a correspondence between motions of bar and body frameworks in 3-space and 6-dimensional scenes over 5-dimensional pictures.     

Decision support models in climate policy

Climate change is considered among the most critical risks that global society faces in this century. So far, climate policy strategies have been evaluated by means of a variety of climate-economy models, or Integrated Assessment Models (IAMs), in the aim of supporting climate-related decision making. However, their inherent complexity, the number and nature of driving assumptions, and usual exclusion of stakeholders from the modelling process raise the issue of the extent to which they can provide fruitful insights for policy makers. Moreover, as with all modelling frameworks, IAMs inevitably fail to incorporate all relevant types of uncertainty and risk when used as stand-alone tools. This exclusion can have a significant impact on the model outcomes, but can be mitigated if experts’ knowledge is elicited in a structured manner and effectively taken into account, towards identifying such factors or reducing respective knowledge gaps. At the same time, a growing number of research publications have been suggesting decision support frameworks for assessing specific aspects in climate policy, based on “bottom-up” approaches and participatory processes. The objective of this paper is to provide a critical review of such frameworks—namely Portfolio Analysis, Multiple Criteria Decision Making and Fuzzy Cognitive Maps—in order to explore their strengths and weaknesses in this area, and propose a new integrative approach, appropriately exploiting blends of these frameworks, to productively complement IAMs, towards enhancing climate policy support.     

Bootstrapping for Penalized Spline Regression

We describe and contrast several different bootstrap procedures for penalized spline smoothers. The bootstrap methods considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered as an estimation technique to find an unknown smooth function. The smooth function is represented in a high-dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrap methods can be implemented under both frameworks, and we show theoretically and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrap ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives.

Datasets and computer code are available in the online supplements.     

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordán, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in \({{\mathbb {R}}}^2\) . The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in \({\mathbb {R}}^3\) when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson’s necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.     

Formal approaches to rule-based systems in medicine: The case of CADIAG-2

There is no established formal framework for expert systems based on weighted IF–THEN rules. We discuss three mathematical models that have been recently proposed by the authors for CADIAG-2—a well-known system of this kind. The three frameworks are based on fuzzy logics, probability theory and possibilistic logic, respectively. CADIAG-2 is used here as a case study to evaluate these frameworks. We point out their use, advantages and disadvantages. In addition, the described models provide insight into various aspects of CADIAG-2.     

Conceptualizing inquiry-based education in mathematics

The terms inquiry-based learning and inquiry-based education have appeared with increasing frequency in educational policy and curriculum documents related to mathematics and science education over the past decade, indicating a major educational trend. We go back to the origin of inquiry as a pedagogical concept in the work of Dewey (e.g. 1916, 1938) to analyse and discuss its migration to science and mathematics education. For conceptualizing inquiry-based mathematics education (IBME) it is important to analyse how this concept resonates with already well-established theoretical frameworks in mathematics education. Six such frameworks are analysed from the perspective of inquiry: the problem-solving tradition, the theory of didactical situations, the realistic mathematics education programme, the mathematical modelling perspective, the anthropological theory of didactics, and the dialogical and critical approach to mathematics education. In an appendix these frameworks are illustrated with paradigmatic examples of teaching activities with inquiry elements. The paper is rounded off with a list of ten concerns for the development and implementation of IBME.     

Convergence and Computational Analyses for Some Variable Target Value and Subgradient Deflection Methods

We consider two variable target value frameworks for solving large-scale nondifferentiable optimization problems. We provide convergence analyses for various combinations of these variable target value frameworks with several direction-finding and step-length strategies including the pure subgradient method, the volume algorithm, the average direction strategy, and a generalized Polyak-Kelley cutting plane method. In addition, we suggest a further enhancement via a projected quadratic-fit line-search whenever any of these algorithmic procedures experiences an improvement in the objective value. Extensive computational results on different classes of problems reveal that these modifications and enhancements significantly improve the effectiveness of the algorithms to solve Lagrangian duals of linear programs, even yielding a favorable comparison against the commercial software CPLEX 8.1.     

Relatively maximal monotone mappings and applications to general inclusions

Based on the relative maximal monotonicity frameworks, the approximation solvability of a general class of variational inclusion problems is explored, while generalizing most of the investigations on weak convergence using the proximal point algorithm in a real Hilbert space setting. Furthermore, the main result has been applied to the context of the relative maximal relaxed monotonicity frameworks for solving a general class of variational inclusion problems. It seems that the obtained results can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution inclusions, and the obtained results can further be applied to the Douglas–Rachford splitting method for finding the zero of the sum of two relatively monotone mappings as well.     

Some aspects of studying an optimization or decision problem in different computational models

In this paper we want to discuss some of the features coming up when analyzing a problem in different complexity theoretic frameworks. The focus will be on two problems. The first is related to mathematical optimization. We consider the quadratic programming problem of minimizing a quadratic polynomial on a polyhedron. We discuss how the complexity of this problem might change if we consider real data together with an algebraic model of computation (the Blum–Shub–Smale model) instead of rational inputs together with the Turing machine model. The results obtained will lead us to the second problem; it deals with the intrinsic structure of complexity classes in models over real- or algebraically closed fields. A classical theorem by Ladner for the Turing model is examined in these different frameworks. Both examples serve well for working out in how far different approaches to the same problem might shed light upon each other. In some cases this will lead to quite diverse results with respect to the different models. On the other hand, for some problems the more general approach can also give a unifying idea why results hold true in several frameworks.The paper is of tutorial character in that it collects some results into the above direction obtained previously.     

Elementary strong maps of graphic matroids

Elementary strong maps of graphic matroids are described in terms of the graph only. An application to rigidity of special bar and joint frameworks is also mentioned.