How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

My Hand or Yours? Markedly Different Sensitivity to Egocentric and Allocentric Views in the Hand Laterality Task

In the hand laterality task participants judge the handedness of visually presented stimuli – images of hands shown in a variety of postures and views - and indicate whether they perceive a right or left hand. The task engages kinaesthetic and sensorimotor processes and is considered a standard example of motor imagery. However, in this study we find that while motor imagery holds across egocentric views of the stimuli (where the hands are likely to be one's own), it does not appear to hold across allocentric views (where the hands are likely to be another person's). First, we find that psychophysical sensitivity, d', is clearly demarcated between egocentric and allocentric views, being high for the former and low for the latter. Secondly, using mixed effects methods to analyse the chronometric data, we find high positive correlation between response times across egocentric views, suggesting a common use of motor imagery across these views. Correlations are, however, considerably lower between egocentric and allocentric views, suggesting a switch from motor imagery across these perspectives. We relate these findings to research showing that the extrastriate body area discriminates egocentric (‘self’) and allocentric (‘other’) views of the human body and of body parts, including hands

Towards a verified compiler prototype for the synchronous language SIGNAL

International audienceSIGNAL belongs to the synchronous languages family which are widely used in the design of safety-critical real-time systems such as avionics, space systems, and nuclear power plants. This paper reports a compiler prototype for SIGNAL. Compared with the existing SIGNAL compiler, we propose a new intermediate representation (named S-CGA, a variant of clocked guarded actions), to integrate more synchronous programs into our compiler prototype in the future. The front-end of the compiler, i.e., the translation from SIGNAL to S-CGA, is presented. As well, the proof of semantics preservation is mechanized in the theorem prover Coq. Moreover, we present the back-end of the compiler, including sequential code generation and multithreaded code generation with time-predictable properties. With the rising importance of multi-core processors in safety-critical embedded systems or cyber-physical systems (CPS), there is a growing need for model-driven generation of multithreaded code and thus mapping on multi-core. We propose a time-predictable multi-core architecture model in architecture analysis and design language (AADL), and map the multi-threaded code to this model

Bioavailability of iodine in the UK-Peak District environment and its human bioaccessibility: an assessment of the causes of historical goitre in this area

Iodine is an essential micronutrient for human health. Its deficiency causes a number of functional and developmental abnormalities such as goitre. The limestone region of Derbyshire, UK was goitre-endemic until it declined from the 1930s and the reason for this has escaped a conclusive explanation. The present study investigates the cause(s) of goitre in the UK-Peak District area through an assessment of iodine in terms of its environmental mobility, bioavailability, uptake into the food chain and human bioaccessibility. The goitre-endemic limestone area is compared with the background millstone grit area of the UK-Peak District. The findings of this study show that ‘total’ environmental iodine is not linked to goitre in the limestone area, but the governing factors include iodine mobility, bioavailability and bioaccessibility. Compared with the millstone grit area, higher soil pH and calcium content of the limestone area restrict iodine mobility in this area, also soil organic carbon in the limestone area is influential in binding the iodine to the soil. Higher calcium content in the limestone area is an important factor in terms of strongly fixing the iodine to the soil. Higher iodine bioaccessibility in the millstone grit than the limestone area suggests that its oral bioaccessibility is restricted in the limestone area. Iodine taken up by plant roots is transported freely into the aerial plant parts in the millstone grit area unlike the limestone area, thus providing higher iodine into the human food chain in the millstone grit area through grazing animals unlike the goitre-prevalent limestone area

Prioritizing micronutrients for the purpose of reviewing their requirements: a protocol developed by EURRECA

Background: The EURRECA (EURopean micronutrient RECommendations Aligned) Network of Excellence (http://www.eurreca.org) is working towards the development of aligned recommendations. A protocol was required to assign resources to those micronutrients for which recommendations are most in need of alignment. Methods: Three important 'a priori' criteria were the basis for ranking micronutrients: (A) the amount of new scientific evidence, particularly from randomized controlled trials; (B) the public health relevance of micronutrients; (C) variations in current micronutrient recommendations. A total of 28 micronutrients were included in the protocol, which was initially undertaken centrally by one person for each of the different population groups defined in EURRECA: infants, children and adolescents, adults, elderly, pregnant and lactating women, and low income and immigrant populations. The results were then reviewed and refined by EURRECA's population group experts. The rankings of the different population groups were combined to give an overall average ranking of micronutrients. Results: The 10 highest ranked micronutrients were vitamin D, iron, folate, vitamin B12, zinc, calcium, vitamin C, selenium, iodine and copper. Conclusions: Micronutrient recommendations should be regularly updated to reflect new scientific nutrition and public health evidence. The strategy of priority setting described in this paper will be a helpful procedure for policy makers and scientific advisory bodies. European Journal of Clinical Nutrition (2010) 64, S19-530; doi:10.1038/ejcn.2010.5

Congenital hypothyroidism

Congenital hypothyroidism (CH) occurs in approximately 1:2,000 to 1:4,000 newborns. The clinical manifestations are often subtle or not present at birth. This likely is due to trans-placental passage of some maternal thyroid hormone, while many infants have some thyroid production of their own. Common symptoms include decreased activity and increased sleep, feeding difficulty, constipation, and prolonged jaundice. On examination, common signs include myxedematous facies, large fontanels, macroglossia, a distended abdomen with umbilical hernia, and hypotonia. CH is classified into permanent and transient forms, which in turn can be divided into primary, secondary, or peripheral etiologies. Thyroid dysgenesis accounts for 85% of permanent, primary CH, while inborn errors of thyroid hormone biosynthesis (dyshormonogeneses) account for 10-15% of cases. Secondary or central CH may occur with isolated TSH deficiency, but more commonly it is associated with congenital hypopitiutarism. Transient CH most commonly occurs in preterm infants born in areas of endemic iodine deficiency. In countries with newborn screening programs in place, infants with CH are diagnosed after detection by screening tests. The diagnosis should be confirmed by finding an elevated serum TSH and low T4 or free T4 level. Other diagnostic tests, such as thyroid radionuclide uptake and scan, thyroid sonography, or serum thyroglobulin determination may help pinpoint the underlying etiology, although treatment may be started without these tests. Levothyroxine is the treatment of choice; the recommended starting dose is 10 to 15 mcg/kg/day. The immediate goals of treatment are to rapidly raise the serum T4 above 130 nmol/L (10 ug/dL) and normalize serum TSH levels. Frequent laboratory monitoring in infancy is essential to ensure optimal neurocognitive outcome. Serum TSH and free T4 should be measured every 1-2 months in the first 6 months of life and every 3-4 months thereafter. In general, the prognosis of infants detected by screening and started on treatment early is excellent, with IQs similar to sibling or classmate controls. Studies show that a lower neurocognitive outcome may occur in those infants started at a later age (> 30 days of age), on lower l-thyroxine doses than currently recommended, and in those infants with more severe hypothyroidism