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Because the modern concept of number emerged within a quadrivium that included music alongside arithmetic, geometry, and astronomy, musical considerations affected mathematical developments. Michael Stifel embedded the then-paradoxical term "irrational numbers" (numerici irrationales) in a musical context (1544), though his philosophical aversion to the "cloud of infinity" surrounding such numbers finally outweighed his musical arguments in their favor. Girolamo Cardano gave the same status to irrational and rational quantities in his algebra (1545), for which his contemporaneous work on music suggested parallels and empirical examples. Nicola Vicentino's attempt to revive ancient "enharmonic" music (1555) required and hence defended the use of "irrational proportions" ( proportiones inrationales) as if they were numbers. These developments emerged in richly interactive social and cultural milieus whose participants interwove musical and mathematical interests so closely that their intense controversies about ancient Grék music had repercussions for mathematics as well. The musical interests of Stifel, Cardano, and Vicentino influenced their respective treatments of "irrational numbers." Practical as well as theoretical music both invited and opened the way for the recognition of a radically new concept of number, even in the téth of paradox. © 2010 by The History of Science Society. All rights reserved.


Hopker J.G.,University of Kent | Jobson S.A.,University of Kent | Pandit J.J.,St Johns College
Anaesthesia | Year: 2011

This article reviews the notion of the 'anaerobic threshold' in the context of cardiopulmonary exercise testing. Primarily, this is a review of the proposed mechanisms underlying the ventilatory and lactate response to incremental exercise, which is important to the clinical interpretation of an exercise test. Since such tests are often conducted for risk stratification before major surgery, a failure to locate or justify the existence of an anaerobic threshold will have some implications for clinical practice. We also consider alternative endpoints within the exercise response that might be better used to indicate a patient's capacity to cope with the metabolic demands encountered both during and following major surgery. © 2011 The Authors.


Grafen A.,St Johns College
Biology and Philosophy | Year: 2014

The broader context for the formal darwinism project established by two of the commentators, in terms of reconciling the Modern Synthesis with Darwinian arguments over design and in terms of links to other types of selection and design, is discussed and welcomed. Some overselling of the project is admitted, in particular of whether it claims to consider all organic design. One important fundamental question raised in two commentaries is flagged but not answered of whether design is rightly represented by an optimisation program, and another from one commentary of whether the coreplicon dissolves in the face of multi-generational imprinting. Calls for the project to be extended to design at levels above and below the individual are considered sympathetically, but judged impractical at the high level of abstraction of the project. All claims of substantive technical error are emphatically rejected. Close technical readings are welcomed that, among other things, represent the project as 'axiomatizing fitness'. The prospects for the project are set out in the light of this highly varied set of commentaries. © 2014 Springer Science+Business Media Dordrecht.


Wynn A.,St Johns College
Complex Analysis and Operator Theory | Year: 2010

Abstract.: In this paper a weighted form of the Weiss conjecture is studied. For certain weights, the conjecture is shown to hold for normal contraction operators related to discrete time linear systems. This is proved by an application of the Carleson measure theorem for weighted Dirichlet spaces. The result for discrete time systems is used to show that a weighted form of the Weiss conjecture holds for normal operators generating bounded C0-semigroups. Previously, weighted admissibility has been characterised for generators of analytic semigroups. No such assumption of analyticity is made here. Additionally, results are presented regarding weighted Carleson measures, fractional powers of normal operators and weighted composition operators. © 2008 Birkhäuser Verlag Basel/Switzerland.


Hulme C.,University College London | Snowling M.J.,University of York | Snowling M.J.,St Johns College
Child Development Perspectives | Year: 2013

The authors review current knowledge about the cognitive processes underlying the early stages of word reading development. Recent findings in a variety of alphabetic languages converge on the conclusion that there are 3 "cognitive foundations" for learning to read: letter-sound knowledge, phonemic awareness, and rapid automatized naming skills. Deficits in each of these skills appear causally related to problems in learning to read, and deficits in letter-sound knowledge and phonemic awareness appear to be remediable by suitable teaching. The authors argue that this evidence has important practical implications for early education and for the diagnosis and treatment of children with reading difficulties. © 2012 The Authors. Child Development Perspectives © 2012 The Society for Research in Child Development.

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