Conflicts in the learning of real numbers and limits

question: "Is 0.999... (nought point nine recurring) equal to one, or just less than one?". Many answers contained infinitesimal concepts: "The same, because the difference between them is infinitely small." " The same, for at infinity it comes so close to one it can be considered the same." "Just less than one, but it is the nearest you can get to one without actually saying it is one." "Just less than one, but the difference between it and one is infinitely small." The majority of students thought that 0.999... was less than one. It may be that a few students had been taught using infinitesimal concepts, or that the phrase “just less than one ” had connotations for the students different from those intended by the questioner; but it seems more likely that the answers represent the students ’ own rationalisations made in an attempt to resolve conflicts inherent in the students ’ previous experience of limiting processes. Some conscious and subconscious conflicts Most of the mathematics met in secondary school consists of sophisticated idea

Concept image and concept definition in mathematics with particular reference to limits and continuity

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition. The development of limits and continuity, as taught in secondary school and university, are considered. Various investigations are reported which demonstrate individual concept images differing from the formal theory and containing factors which cause cognitive conflict

Symmetry of Magnetically Ordered Quasicrystals

The notion of magnetic symmetry is reexamined in light of the recent observation of long range magnetic order in icosahedral quasicrystals [Charrier et al., Phys. Rev. Lett. 78, 4637 (1997)]. The relation between the symmetry of a magnetically-ordered (periodic or quasiperiodic) crystal, given in terms of a ``spin space group,'' and its neutron diffraction diagram is established. In doing so, an outline of a symmetry classification scheme for magnetically ordered quasiperiodic crystals is provided. Predictions are given for the expected diffraction patterns of magnetically ordered icosahedral crystals, provided their symmetry is well described by icosahedral spin space groups.Comment: 5 pages. Accepted for publication in Phys. Rev. Letter

Diffractive point sets with entropy

After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.Comment: 25 pages; dedicated to Hans-Ude Nissen on the occasion of his 65th birthday; final version, some minor addition

Neutrophils in cancer: neutral no more

Neutrophils are indispensable antagonists of microbial infection and facilitators of wound healing. In the cancer setting, a newfound appreciation for neutrophils has come into view. The traditionally held belief that neutrophils are inert bystanders is being challenged by the recent literature. Emerging evidence indicates that tumours manipulate neutrophils, sometimes early in their differentiation process, to create diverse phenotypic and functional polarization states able to alter tumour behaviour. In this Review, we discuss the involvement of neutrophils in cancer initiation and progression, and their potential as clinical biomarkers and therapeutic targets

Factors affecting patterns of tick parasitism on forest rodents in tick-borne encephalitis risk areas, Germany

Identifying factors affecting individual vector burdens is essential for understanding infectious disease systems. Drawing upon data of a rodent monitoring programme conducted in nine different forest patches in southern Hesse, Germany, we developed models which predict tick (Ixodes spp. and Dermacentor spp.) burdens on two rodent species Apodemus flavicollis and Myodes glareolus. Models for the two rodent species were broadly similar but differed in some aspects. Patterns of Ixodes spp. burdens were influenced by extrinsic factors such as season, unexplained spatial variation (both species), relative humidity and vegetation cover (A. flavicollis). We found support for the ‘body mass’ (tick burdens increase with body mass/age) and for the ‘dilution’ hypothesis (tick burdens decline with increasing rodent densities) and little support for the ‘sex-bias’ hypothesis (both species). Surprisingly, roe deer densities were not correlated with larvae counts on rodents. Factors influencing the mean burden did not significantly explain the observed dispersion of tick counts. Co-feeding aggregations, which are essential for tick-borne disease transmission, were mainly found in A. flavicollis of high body mass trapped in areas with fast increase in spring temperatures. Locally, Dermacentor spp. appears to be an important parasite on A. flavicollis and M. glareolus. Dermacentor spp. was rather confined to areas with higher average temperatures during the vegetation period. Nymphs of Dermacentor spp. mainly fed on M. glareolus and were seldom found on A. flavicollis. Whereas Ixodes spp. is the dominant tick genus in woodlands of our study area, the distribution and epidemiological role of Dermacentor spp. should be monitored closely

Between history and values: A study on the nature of interpretation in international law

My thesis discusses the place of evaluative judgements in the interpretation of general international law. It concentrates on two questions. First, whether it is possible to interpret international legal practices without making an evaluative judgement about the point or value that provides the best justification of these practices. Second, whether the use of evaluative judgements in international legal interpretation threatens to undermine the objectivity of international law, the neutrality of international lawyers or the consensual and voluntary basis of the international legal system. I answer both questions in the negative. As regards the first, I argue that international legal practice has an interpretive structure, which combines appeals to the history of international practice with appeals to the principles and values that these practices are best understood as promoting. This interpretive structure is apparent not only in the claims of international lawyers about particular rules of international law (here I use the rule of estoppel as an example) but also in the most basic intuitions of international theorists about the theory and sources of general international law. I then argue that some popular concerns to the effect that the exercise of evaluation in the interpretation of international law will undermine the coherence or the usefulness of the discipline are generally unwarranted. The fact that international legal practice has an interpretive structure does not entail that propositions of international law are only subjectively true, that the interpreter enjoys license to manipulate their meaning for self-serving purposes, or that international law will collapse under the weight of irresolvable disagreements, divisions and conflicts about its proper interpretation