Exact arithmetic on the Stern–Brocot tree

AbstractIn this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials

«La plus grande audace à notre époque: être simple». Simplicity, Subversion and Intimations of the Surreal in Francis Poulenc’s Early Collaborations with Jean Cocteau

This article discusses the concept of simplicity in Cocardes and other early works by Poulenc created in collaboration with Jean Cocteau. Drawing together strands ranging chronologically and stylistically from the ballet Parade through Cocteau’s championing of Satie’s music in Le coq et l’arlequin and the influence of Tzara and dadaism to the eventual parting of the ways between Dada and Surrealism, it suggests a complex and subversive role for simplicity in this context and finds traces of a dadaist and even a proto-surrealist spirit in the way it is deployed in these compositions.«La plus grande audace à notre époque: être simple». Semplicità, sovversione e accenni del surreale nelle prime collaborazioni di Francis Poulenc con Jean CocteauQuesto articolo tratta il concetto di semplicità in Cocardes e in altri lavori giovanili di Poulenc creati in collaborazione con Jean Cocteau. Mettendo insieme elementi che vanno, cronologicamente e stilisticamente, dal balletto Parade, attraverso il sostegno di Cocteau alla musica di Satie in Le coq et l’arlequin e l’influenza di Tzara e del dadaismo, fino alla finale separazione delle strade di Dada e del Surrealismo, suggerisce un ruolo complesso e sovversivo della semplicità in questo contesto e trova tracce di uno spirito dadaista e persino proto-surrealista nel modo in cui esso è sviluppato in queste composizioni.This article discusses the concept of simplicity in Cocardes and other early works by Poulenc created in collaboration with Jean Cocteau. Drawing together strands ranging chronologically and stylistically from the ballet Parade through Cocteau’s championing of Satie’s music in Le coq et l’arlequin and the influence of Tzara and dadaism to the eventual parting of the ways between Dada and Surrealism, it suggests a complex and subversive role for simplicity in this context and finds traces of a dadaist and even a proto-surrealist spirit in the way it is deployed in these compositions.«La plus grande audace à notre époque: être simple». Semplicità, sovversione e accenni del surreale nelle prime collaborazioni di Francis Poulenc con Jean CocteauQuesto articolo tratta il concetto di semplicità in Cocardes e in altri lavori giovanili di Poulenc creati in collaborazione con Jean Cocteau. Mettendo insieme elementi che vanno, cronologicamente e stilisticamente, dal balletto Parade, attraverso il sostegno di Cocteau alla musica di Satie in Le coq et l’arlequin e l’influenza di Tzara e del dadaismo, fino alla finale separazione delle strade di Dada e del Surrealismo, suggerisce un ruolo complesso e sovversivo della semplicità in questo contesto e trova tracce di uno spirito dadaista e persino proto-surrealista nel modo in cui esso è sviluppato in queste composizioni

Resisting in France and la vie inventée

The daily experience of resistance in occupied France has often been missing from accounts of les années noires. Whether concerned with the deeds of prominent resisters or with the deconstruction of national myths, history has often obscured the experiences of the majority of the significant minority who opted to rebel against oppression. The first two years of the Occupation are often overshadowed by the move towards a unified movement and the increasingly combative stance of the Resistance of the following years. This may be partly related to the difficulty in placing such disparate realities into a coherent methodological framework. Equally, an analysis of events that possessed a surreal and almost dreamlike quality by those that witnessed them may have discouraged attempts to gain a deeper awareness of the phenomenon of resistance. In some respects, it did occupy a different sphere of reality for those that chose not to obey the armistice could be considered marginal in their behaviour and their memory remained so in post-war France as the demands of national reconstruction produced a dominant representation of the period which obscured the experience of the individual. This paper seeks to explore this sub-reality through an analysis of la vie inventée and its manifestation within the creation of an ésprit de résistance, the transmission of this consciousness and the inversion of the hegemony of the Vichy régime. Finally, it will question whether this notion was born from or conversely a prerequisite to La Résistance

Verification of Dependable Software using SPARK and Isabelle

We present a link between the interactive proof assistant Isabelle/HOL and the SPARK/Ada tool suite for the verification of high-integrity software. Using this link, we can tackle verification problems that are beyond reach of the proof tools currently available for Spark. To demonstrate that our methodology is suitable for real-world applications, we show how it can be used to verify an efficient library for big numbers. This library is then used as a basis for an implementation of the RSA public-key encryption algorithm in SPARK/Ada

2020: murburn concept heralds a new era in cellular bioenergetics

Cellular bioenergetics has been interpreted for several decades using the Keilin-Mitchell-Boyer (KMB) model of oxidative phosphorylation (OxPhos), and for understanding/managing of the pertinent mitochondrial pathophysiological states. Although decades of research had revealed many faulty chemico-physical aspects of KMB perspective, recent critical insights from our group’s writings have sufficiently brought out the errors in the KMB model, rendering it obsolete/redundant. The murburn model proposed in lieu is a compelling alternative for explaining OxPhos because it reasons several facets of mitochondrial structure-function correlations, reaction chemistry and thermodynamics. However, the mitochondrial research community appears to be recalcitrant, and continues to follow the erstwhile erroneous ideas and not take cognizance of the new insights. Hence, we deemed it opportune to make a clarion call for a jettisoning of the superseded terminologies (or keywords) and concepts routinely used by researchers in this field. First, we present a statistical perspective of the usage of these terms in the past and recent times, to support our claims and call. Then, we articulate simplified arguments why the key elements/terms of the KMB model like “electron-transfer/electron-transport/respiratory chain”, “mitochondrial proton pumps”, “mitochondrial membrane po-tential”, “chemiosmosis”, “proton motive force” and “rotary ATP synthase/synthesis” violate scientific/semantic logic. Finally, we conclude with summative statements projecting the importance of our claims and call

Transporting Functions across Ornaments

Programming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of data-types: we can finally write correct-by-construction software. However, this extreme accuracy is also a curse: a data-type is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any effort of code reuse among similarly structured data. In this paper, we exorcise our data-types by adapting the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornament, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: the user can ask the definition of addition to be lifted to lists and she will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in a type theory with a universe of data-types and all our constructions have been implemented as generic programs, requiring no extension to the type theory

Aspects of surrealism in the work of Jean Cocteau

The work of Jean Cocteau reveals connections, similarities and differences between him and the writers of the surrealist movement. In order to appreciate the links it is also necessary to examine the principles of Surrealism to determine the extent to which they have similar origins to some of Cocteau's own ideas.This line of inquiry leads to an examination of the part played by the work of Freud and Jung in inspiring both Surrealism and Cocteau. To a certain degree one is lead to question the association between Freud and Surrealism which has often been taken for granted and to look for the origins of surrealistic thought in more specifically French sources to which Freud also had access. Whilst it is difficult to bring about a rapprochement between Cocteau and Freud, there is a much smaller problem in comparing the work of Jung and that of Cocteau. There are striking similarities which indicate not only a divergence of thought between Cocteau and the surrealists but which also tempt one to extrapolate a direct link between Cocteau and Jung for which there is virtually no direct evidence. What is achieved in comparing the two is a greater understanding of the creative method of Cocteau, of the forces which drove him, and of his basic position as a child of the 20th century, yet as a poet of all ages. One begins also to have a clearer vision of the reasons which underlie his all important interest in mythology as a source of pure emotion and distilled poetic essence. For personal rather than artistic reasons a close rapport between Cocteau and the surrealist group is unthinkable as well as generally known, so that there is an enhanced interest not only in a direct comparison but also in comparing Cocteau with artists and poets who worked close to official movement but were not, at least for very long, part of it. Garcia Lorca is a Spanish writer in this position whose ideas and background so closely,resemble Cocteau's that it is almost surprising to find that he was at least tolerated if not completely accepted by the Surrealists; his friends Dalí and and Bunuel even joined the group formally. On the other hand Cocteau's proclaimed admiration for Garcia Lorca indicated at least some feeling in not being able to participate directly in the Surrealist experience. Comparing Cocteau with Lorca necessitates an examination of the creeds and ideals of them both, highlighting aspects of poetic power and creativity in the process.It is hoped to place in the context of 20th century thought the work of both Cocteau and the Surrealists. A continuity between the second half of the nineteenth century and the twentieth century should also be established and the manner in which the First World War acts as a watershed made clear. From the study Cocteau emerges as a more consistent and deeper thinker than he is often considered. The parallels found in the work which he presented in a variety of different artistic fields coupled with the overpowering sense of mission which begins to appear, dispel for ever the myths of the careless and carefree casual adolescent dilettante and reveal instead a conscious artist, a thinking poet, a careful craftsman and a profoundly proud human figure wrought with deep seated anxieties often masqued with flippancy. Undeniably however, consciously or unconsciously, whether or not the Surrealist liked the idea, there was an affinity between them and Cocteau which was sometimes a very close link and at others flared up into an open hostility which at least indicated that they were working in the same areas.Since it was the fashion at the time to accept the view of Freud as a scientist and a medical practitioner in the field of psychiatry, a view which he himself insisted upon, it has been felt justifiable to accept it, although nowadays he is partially discredited. The concept of the subconscious is also not considered favourably although it seemed real to Freud, Jung, the Surrealists and to Cocteau. Consequently it is desirable to work within the parameters of their imagination rather than to take the stance of modern behaviourist psychiatrists whose ideas are irrelevant to the literary uses made of the work of Freud and Jung

Large and Infinitary Quotient Inductive-Inductive Types

Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures