The Projective Line Over the Finite Quotient Ring GF(2)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement I. Theoretical Background

The paper deals with the projective line over the finite factor ring $R\_{\clubsuit} \equiv$ GF(2)[$x$]/$$. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As $R\_{\clubsuit}$ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. There are nine neighbour points to any point of the line, forming three disjoint families under the reduction modulo either of two maximal ideals of the ring. Two of the families contain four points each and they swap their roles when switching from one ideal to the other; the points of the one family merge with (the image of) the point in question, while the points of the other family go in pairs into the remaining two points of the associated ordinary projective line of order two. The single point of the remaining family is sent to the reference point under both the mappings and its existence stems from a non-trivial character of the Jacobson radical, ${\cal J}\_{\clubsuit}$, of the ring. The factor ring $\widetilde{R}\_{\clubsuit} \equiv R\_{\clubsuit}/ {\cal J}\_{\clubsuit}$ is isomorphic to GF(2) $\otimes$ GF(2). The projective line over $\widetilde{R}\_{\clubsuit}$ features nine points, each of them being surrounded by four neighbour and the same number of distant points, and any two distant points share two neighbours. These remarkable ring geometries are surmised to be of relevance for modelling entangled qubit states, to be discussed in detail in Part II of the paper.Comment: 8 pages, 2 figure

Collaboration between Mathematics and Mathematics Education

Abstract Our chapter is in four sections. Michèle Artigue tells the story of her transition from mathematical logic to mathematics education and of collaborations at a wide variety of institutional levels. Günter Törner gives a history of collaboration between mathematics and mathematics education in Germany along with a list of recommendations to foster collaboration. Ehud de Shalit shares lessons learned from personal experiences collaborating in the production of a math fair and in the design of a mathematics education major. Pat Thompson tells of several collaborative efforts at his home institution and examines ways that mathematics education contributed mathematically to them. A concluding section provides a reflection on our charge -structural and cultural issues involved in collaborations between mathematics and mathematics education

Area Disease Estimation Based on Sentinel Hospital Records

BACKGROUND: Population health attributes (such as disease incidence and prevalence) are often estimated using sentinel hospital records, which are subject to multiple sources of uncertainty. When applied to these health attributes, commonly used biased estimation techniques can lead to false conclusions and ineffective disease intervention and control. Although some estimators can account for measurement error (in the form of white noise, usually after de-trending), most mainstream health statistics techniques cannot generate unbiased and minimum error variance estimates when the available data are biased. METHODS AND FINDINGS: A new technique, called the Biased Sample Hospital-based Area Disease Estimation (B-SHADE), is introduced that generates space-time population disease estimates using biased hospital records. The effectiveness of the technique is empirically evaluated in terms of hospital records of disease incidence (for hand-foot-mouth disease and fever syndrome cases) in Shanghai (China) during a two-year period. The B-SHADE technique uses a weighted summation of sentinel hospital records to derive unbiased and minimum error variance estimates of area incidence. The calculation of these weights is the outcome of a process that combines: the available space-time information; a rigorous assessment of both, the horizontal relationships between hospital records and the vertical links between each hospital's records and the overall disease situation in the region. In this way, the representativeness of the sentinel hospital records was improved, the possible biases of these records were corrected, and the generated area incidence estimates were best linear unbiased estimates (BLUE). Using the same hospital records, the performance of the B-SHADE technique was compared against two mainstream estimators. CONCLUSIONS: The B-SHADE technique involves a hospital network-based model that blends the optimal estimation features of the Block Kriging method and the sample bias correction efficiency of the ratio estimator method. In this way, B-SHADE can overcome the limitations of both methods: Block Kriging's inadequacy concerning the correction of sample bias and spatial clustering; and the ratio estimator's limitation as regards error minimization. The generality of the B-SHADE technique is further demonstrated by the fact that it reduces to Block Kriging in the case of unbiased samples; to ratio estimator if there is no correlation between hospitals; and to simple statistic if the hospital records are neither biased nor space-time correlated. In addition to the theoretical advantages of the B-SHADE technique over the two other methods above, two real world case studies (hand-foot-mouth disease and fever syndrome cases) demonstrated its empirical superiority, as well

Identifying work ability promoting factors for home care aides and assistant nurses

<p>Abstract</p> <p>Background</p> <p>In workplace health promotion, all potential resources needs to be taken into consideration, not only factors relating to the absence of injury and the physical health of the workers, but also psychological aspects. A dynamic balance between the resources of the individual employees and the demands of work is an important prerequisite. In the home care services, there is a noticeable trend towards increased psychosocial strain on employees at work. There are a high frequency of work-related musculoskeletal disorders and injuries, and a low prevalence of sustainable work ability. The aim of this research was to identify factors promoting work ability and self-efficacy in care aides and assistant nurses within home care services.</p> <p>Methods</p> <p>This study is based on cross-sectional data collected in a municipality in northern Sweden. Care aides (n = 58) and assistant nurses (n = 79) replied to a self-administered questionnaire (response rate 46%). Hierarchical multiple regression analyses were performed to assess the influence of several independent variables on self-efficacy (model 1) and work ability (model 2) for care aides and assistant nurses separately.</p> <p>Results</p> <p>Perceptions of personal safety, self-efficacy and musculoskeletal wellbeing contributed to work ability for assistant nurses (R²adj of 0.36, <it>p </it>< 0.001), while for care aides, the safety climate, seniority and age contributed to work ability (R²adj of 0.29, <it>p </it>= 0.001). Self-efficacy was associated with the safety climate and the physical demands of the job in both professions (R²adj of 0.24, <it>p </it>= 0.003 for care aides), and also by sex and age for the assistant nurses (R²adj of 0.31, <it>p </it>< 0.001).</p> <p>Conclusions</p> <p>The intermediate factors contributed differently to work ability in the two professions. Self-efficacy, personal safety and musculoskeletal wellbeing were important for the assistant nurses, while the work ability of the care aides was associated with the safety climate, but also with the non-changeable factors age and seniority. All these factors are important to acknowledge in practice and in further research. Proactive workplace interventions need to focus on potentially modifiable factors such as self-efficacy, safety climate, physical job demands and musculoskeletal wellbeing.</p

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions

Beliefs and beyond : affect and the teaching and learning of mathematics

The importance of beliefs for the teaching and learning of mathematics is widely recognized among mathematics educators. In this special issue, we explicitly address what we call "beliefs and beyond" to indicate the larger field surrounding beliefs in mathematics education. This is done to broaden the discussion to related concepts (which may not originate in mathematics education) and to consider the interconnectedness of concepts. In particular, we present some new developments at the conceptual level, address different approaches to investigate beliefs, highlight the role of student beliefs in problem-solving activities, and discuss teacher beliefs and their significance for professional development. One specific intention is to consider expertise from colleagues in the fields of educational research and psychology, side by side with perspectives provided by researchers from mathematics education

Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts

In this paper we provide a first overview of the landscape with respect to calculus teaching in European classrooms, an area where research is very limited. In particular through a small expert-based survey and a literature review, we trace the development of calculus teaching at schools in a number of European countries and identify commonalities and differences. In the current curriculum developments, we notice a reduction in the content of calculus and a more informal approach. The use of digital tools has started to be integrated in calculus teaching in most countries. However in some nations, teaching of calculus in the classroom is rather traditional, focusing on procedural aspects of knowledge. Moreover, in cases where more informal and conceptual teaching approaches are used in the classroom, often contradictions seem to exist with other contextual matters such as examination requirements. Finally, we discuss the future of calculus teaching in Europe. © 2014, FIZ Karlsruhe

Current topics within international mathematics education research - Editorial

Törner G, Gross C, Kleine M, Sriraman B. Current topics within international mathematics education research - Editorial. Zentralblatt für Didaktik der Mathematik (ZDM). 2005;37(3):129