Construction of the transcomplex numbers from the complex numbers

A geometrical construction of the transcomplex numbers was given elsewhere. Here we simplify the transcomplex plane and construct the set of transcomplex numbers from the set of complex numbers. Thus transcomplex numbers and their arithmetic arise as consequences of their construction, not by an axiomatic development. This simplifes transcom- plex arithmetic, compared to the previous treatment, but retains totality so that every arithmetical operation can be applied to any transcomplex number(s) such that the result is a transcomplex number. Our proof establishes the consistency of transcomplex and transreal arithmetic and establishes the expected containment relationships amongst transcomplex, complex, transreal and real numbers. We discuss some of the advantages the transarithmetics have over their partial counterparts.

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 limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

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.

Transreal limits and elementary functions

We extend all elementary functions from the real to the transreal domain so that they are defined on division by zero. Our method applies to a much wider class of functions so may be of general interest.

Transreal proof of the existence of universal possible worlds

Transreal arithmetic is total, in the sense that the fundamental operations of addition, subtraction, multiplication and division can be applied to any transreal numbers with the result being a transreal number [1]. In particular division by zero is allowed. It is proved, in [3], that transreal arithmetic is consistent and contains real arithmetic. The entire set of transreal numbers is a total semantics that models all of the semantic values, that is truth values, commonly used in logics, such as the classical, dialetheaic, fuzzy and gap values [2]. By virtue of the totality of transreal arithmetic, these logics can be implemented using total, arithmetical functions, specifically operators, whose domain and counterdomain is the entire set of transreal numbers

Transreal logical space of all propositions

Transreal numbers provide a total semantics containing classical truth values, dialetheaic, fuzzy and gap values. A paraconsistent Sheffer Stroke generalises all classical logics to a paraconsistent form. We introduce logical spaces of all possible worlds and all propositions. We operate on a proposition, in all possible worlds, at the same time. We define logical transformations, possibility and necessity relations, in proposition space, and give a criterion to determine whether a proposition is classical. We show that proofs, based on the conditional, infer gaps only from gaps and that negative and positive infinity operate as bottom and top values.

The politics of slowness and the traps of modernity

This chapter re-evaluates the diachronic, evolutionist model that establishes the Second World War as a watershed between classical and modern cinemas, and ‘modernity’ as the political project of ‘slow cinema’. I will start by historicising the connection between cinematic speed and modernity, going on to survey the veritable obsession with the modern that continues to beset film studies despite the vagueness and contradictions inherent in the term. I will then attempt to clarify what is really at stake within the modern-classical debate by analysing two canonical examples of Japanese cinema, drawn from the geidomono genre (films on the lives of theatre actors), Kenji Mizoguchi’s Story of the Late Chrysanthemums (Zangiku monogatari, 1939) and Yasujiro Ozu’s Floating Weeds (Ukigusa, 1954), with a view to investigating the role of the long take or, conversely, classical editing, in the production or otherwise of a supposed ‘slow modernity’. By resorting to Ozu and Mizoguchi, I hope to demonstrate that the best narrative films in the world have always combined a ‘classical’ quest for perfection with the ‘modern’ doubt of its existence, hence the futility of classifying cinema in general according to an evolutionary and Eurocentric model based on the classical-modern binary. Rather than on a confusing politics of the modern, I will draw on Bazin’s prophetic insight of ‘impure cinema’, a concept he forged in defence of literary and theatrical screen adaptations. Anticipating by more than half a century the media convergence on which the near totality of our audiovisual experience is currently based, ‘impure cinema’ will give me the opportunity to focus on the confluence of film and theatre in these Mizoguchi and Ozu films as the site of a productive crisis where established genres dissolve into self-reflexive stasis, ambiguity of expression and the revelation of the reality of the film medium, all of which, I argue, are more reliable indicators of a film’s political programme than historical teleology. At the end of the journey, some answers may emerge to whether the combination of the long take and the long shot are sufficient to account for a film’s ‘slowness’ and whether ‘slow’ is indeed the best concept to signify resistance to the destructive pace of capitalism.

Analysis of phosphorus use efficiency traits in Coffea genotypes reveals Coffea arabica and Coffea canephora have contrasting phosphorus uptake and utilization efficiencies

Background and Aims: Phosphate (Pi) is one of the most limiting nutrients for agricultural production in Brazilian soils due to low soil Pi concentrations and rapid fixation of fertilizer Pi by adsorption to oxidic minerals and/or precipitation by iron and aluminum ions. The objectives of this study were to quantify phosphorus (P) uptake and use efficiency in cultivars of the species Coffea arabica L. and Coffea canephora L., and group them in terms of efficiency and response to Pi availability. Methods: Plants of 21 cultivars of C. arabica and four cultivars of C. canephora were grown under contrasting soil Pi availabilities. Biomass accumulation, tissue P concentration and accumulation and efficiency indices for P use were measured. Key Results: Coffee plant growth was significantly reduced under low Pi availability, and P concentration was higher in cultivars of C. canephora. The young leaves accumulated more P than any other tissue. The cultivars of C. canephora had a higher root/shoot ratio and were significantly more efficient in P uptake, while the cultivars of C. arabica were more efficient in P utilization. Agronomic P use efficiency varied among coffee cultivars and E16 Shoa, E22 Sidamo, Iêmen and Acaiá cultivars were classified as the most efficient and responsive to Pi supply. A positive correlation between P uptake efficiency and root to shoot ratio was observed across all cultivars at low Pi supply. These data identify Coffea genotypes better adapted to low soil Pi availabilities, and the traits that contribute to improved P uptake and use efficiency. These data could be used to select current genotypes with improved P uptake or utilization efficiencies for use on soils with low Pi availability and also provide potential breeding material and targets for breeding new cultivars better adapted to the low Pi status of Brazilian soils. This could ultimately reduce the use of Pi fertilizers in tropical soils, and contribute to more sustainable coffee production.