Transdifferential and transintegral calculus

The set of transreal numbers is a superset of the real numbers. It totalises real arithmetic by defining division by zero in terms of three def- inite, non-finite numbers: positive infinity, negative infinity and nullity. Elsewhere, in this proceedings, we extended continuity and limits from the real domain to the transreal domain, here we extended the real derivative to the transreal derivative. This continues to demonstrate that transreal analysis contains real analysis and operates at singularities where real analysis fails. Hence computer programs that rely on computing deriva- tives { such as those used in scientific, engineering and financial applica- tions { are extended to operate at singularities where they currently fail. This promises to make software, that computes derivatives, both more competent and more reliable. We also extended the integration of absolutely convergent functions from the real domain to the transreal domain.

Transreal calculus

Transreal arithmetic totalises real arithmetic by defining division by zero in terms of three definite, non-finite numbers: positive infinity, negative infinity and nullity. We describe the transreal tangent function and extend continuity and limits from the real domain to the transreal domain. With this preparation, we extend the real derivative to the transreal derivative and extend proper integration from the real domain to the transreal domain. Further, we extend improper integration of absolutely convergent functions from the real domain to the transreal domain. This demonstrates that transreal calculus contains real calculus and operates at singularities where real calculus fails.

Understanding consumer evaluations of personalised nutrition services in terms of the privacy calculus: a qualitative study

Background: Personalised nutrition (PN) may provide major health benefits to consumers. A potential barrier to the uptake of PN is consumers’ reluctance to disclose sensitive information upon which PN is based. This study adopts the privacy calculus to explore how PN service attributes contribute to consumers’ privacy risk and personalisation benefit perceptions. Methods: Sixteen focus groups (n = 124) were held in 8 EU countries and discussed 9 PN services that differed in terms of personal information, communication channel, service provider, advice justification, scope, frequency, and customer lock-in. Transcripts were content analysed. Results: The personal information that underpinned PN contributed to both privacy risk perception and personalisation benefit perception. Disclosing information face-to-face mitigated the perception of privacy risk and amplified the perception of personalisation benefit. PN provided by a qualified expert and justified by scientific evidence increased participants’ value perception. Enhancing convenience, offering regular face-to face support, and employing customer lock-in strategies were perceived as beneficial. Conclusion: This study suggests that to encourage consumer adoption, PN has to account for face-to-face communication, expert advice providers, support, a lifestyle-change focus, and customised offers. The results provide an initial insight into service attributes that influence consumer adoption of PN.

Evaluating microfossil content of dental calculus from Brazilian sambaquis

To date, limited numbers of dental calculus samples have been analyzed by researchers in diverse parts of the world. The combined analyses of these have provided some general guidelines for the analysis of calculus that is non-destructive to archaeological teeth. There is still a need for a quantitative study of large numbers of calculus samples to establish protocols, assess the level of contamination, evaluate the quantity of microfossils in dental calculus, and to compare analysis results with the literature concerning the biology of calculus formation. We analyzed dental calculus from 53 teeth from four Brazilian sambaquis. Sambaquis are the shell-mounds that were established prehistorically along the Brazilian coast. The analysis of sambaqui dental calculi shows that there are relatively high concentrations of microfossils (phytoliths and starch), mineral fragments, and charcoal in dental calculus. Mineral fragments and charcoal are possibly contaminants. The largest dental calculi have the lowest concentrations of microfossils. Biologically, this is explained by individual variation in calculus formation between people. Importantly, starch is ubiquitous in dental calculus. The starch and phytoliths show that certainly Dioscorea (yam) and Araucaria angustifolia (Parana pine) were eaten by sambaqui people. Araceae (arum family), Ipomoea batatas (sweet potato) and Zea mays (maize) were probably in their diet. (C) 2009 Elsevier Ltd. All rights reserved.

Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.