A heuristic comparison is made of relativistic and non-relativistic quantum theory. To this end the Segal approach is described for the non-specialist. The significance of antimatter to the local and microcausal properties of the fields is laid bare. The fundamental difference between relativistic and non-relativistic (complex) fields is traced to the existence of two kinds of complex numbers in the relativistic case. Their relation to covariant and Newton-Wigner locality is formulated.

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter.

There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There (...) seems to be always a background and a context that we rely upon. Thus mathematicians naturally make use of Kantian intuition and references fixed by names and denotations. I argue that such features cannot be avoided. (shrink)

Robert Brandom; XII*—The Significance of Complex Numbers for Frege's Philosophy of Mathematics1, Proceedings of the Aristotelian Society, Volume 96, Issue 1, 1.

The development and applicability of complex numbers is often cited in defence of the formalist philosophy of mathematics. This view is rejected through an examination of hamilton's development of the notion of complex numbers as ordered pairs of reals, And his later development of the quaternion theory, Which subsequently formed the basis of vector analysis. Formalism, By protecting informal assumptions from critical scrutiny, Constrained rather than encouraged the development of mathematics.

In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke’s Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions (...) that is both necessary and sufficient for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters. (shrink)

A short note on why we use complex numbers in physics.

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was not misleading and was often key to important developments. Thus the geometrization of complex (...) class='Hi'>numbers was essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries. (shrink)

We consider two theories of"bad fields" constructed by B.Poizat using Hrushovski's amalgamation and show that these theories have natural models representable as the field of complex numbers with a distinguished subset given as a union of countably many real analytic curves. One of the two examples is based on the complex exponentiation and the proof assumes Schanuel's conjecture.

In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and (...) class='Hi'>complex numbers in _Grundlagen_ is modelled on his definitional introduction of cardinal numbers in two steps, tentatively via a contextual definition and finally and definitively via an explicit definition. I argue that the strategy leaves a few important questions open, in particular one relating to the status of the envisioned abstraction principles for the real and complex numbers and another concerning the proper handling of cross-sortal identity claims. (shrink)

A transcription and translation of Hermann Hankel's 1867 Vorlesungen über die complexen Zahlen und ihre Functionen, I. Theil: Theorie der Complexen Zahlensysteme, a textbook on complex analysis that played an important role in the transition to modern mathematics in nineteenth century Germany.

This is a bugfix release to correct some errors in.zenodo.json which prevented archiving of 1.0. -/- Version 1.0: This first release of the translation contains: -/- * a full transcription of the Preface and Chapter 1 * translations of select passages in these sections * a full transcription and translation of the title pages and the table of contents through chapter 3 -/- This release was focused on working out (most of) the issues with the markup of a parallel translation (...) and with the build. Subsequent versions will contain more of the text and be more focused on translation! (shrink)

We study some representations of real numbers. We compare these representations, on the one hand from the viewpoint of recursive functionals, and of complexity on the other hand.The impossibility of obtaining some functions as recursive functionals is, in general, easy. This impossibility may often be explicited in terms of complexity: - existence of a sequence of low complexity whose image is not a recursive sequence, - existence of objects of low complexity but whose images have arbitrarily high time- complexity (...) .Moreover, some representations of real numbers that are equivalent from the viewpoint of recursive functionals, are very distinct from the viewpoint of complexity.We make a particular study of representations via continued fractions . We precise exactly what part of information available in the x's dfc is equivalent to the information available in its Dedekinds cut. We show that the sum of two reals whose dfcs are polynomial-time computable may be a real whose dfc has time complexity arbitrarily high.This work confirms that the unique representation of real numbers suitable for the ordinary calculus is via explicit Cauchy sequences of rationals.RésuméNous étudions différentes manières de présenter les nombres réels. Nous comparons ces présentations du point de vue des fonctionnelles récursives d'une part, et de celui des classes de complexité d'autre part.L'impossibilité d'obtenir certaines fonctions sous forme de fonctionnelles récursives est en général facile à établir. Cette impossiblité peut souvent être explicitée en termes de complexité: - il existe une suite de faible complexité dont l'image est une suite non récursive, - il existe des objets de faible complexité mais dont les images sont des objets de complexité arbitrairement grande .En outre, certaines présentations des réels équivalentes du point de vue des fonctionnelles récursives se distinguent nettement du point de vue de la complexité.Nous faisons une étude particulière concernant les développements en fraction continue . Nous précisions exactement quelle est la partie de l'information disponible dans le dfc d'un réel x qui équivaut à l'information disponible dans sa coupure de Dedekind. Nous montrons également que la somme de deux réels dont le dfc est calculable en temps polynomial peut être un réel dont le dfc est de complexité arbitrairement grande.Ce travail confirme que seule une présentation des réels via des suites de rationnels explicitement de Cauchy est adaptée aux calculs avec les réels. (shrink)

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might (...) have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow. Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths. (shrink)

DNA–protein interactions are fundamental to many biological processes, including the regulation of gene expression. Determining the binding affinities of transcription factors (TFs) to different DNA sequences allows the quantitative modeling of transcriptional regulatory networks and has been a significant technical challenge in molecular biology for many years. A recent paper by Maerkl and Quake1 demonstrated the use of microfluidic technology for the analysis of DNA–protein interactions. An array of short DNA sequences was spotted onto a glass slide, which was then (...) covered with a microfluidic device allowing each spot to be within a chamber into which the flow of materials was controlled by valves. By trapping the DNA–protein complexes on the surface and measuring their concentrations microscopically, they could determine the binding affinity to a large number of DNA sequences that were varied systematically. They studied four TFs from the basic helix–loop–helix family of proteins, all of which bind to E‐box sites with the consensus CAnnTG (where “n” can be any base), and showed that variations in affinity for different sites allows each TF to regulate different genes. BioEssays 29:717–721, 2007. © 2007 Wiley Periodicals, Inc. (shrink)

The following result is an approximation to the answer of the question of Kokorin about decidability of a quantifier-free theory of field of rational numbers. Let Q0 be a subset of the set of all rational numbers which contains integers 1 and −1. Let be a set containing Q0 and closed by the functions of addition, subtraction and multiplication. For example coincides with Q0 if Q0 is the set of all binary rational numbers or the set of (...) all decimal rational numbers. It is clear that , where Z is the set of all integers and Q is the set of all rational numbers. A negationless theory uses only conjunction and disjunction as logical connectives. Let T be a quantifier-free negationless elementary theory of signature , where Pol is the list of all polynomials with rational coefficients from Q0 in which exponent of the polynomials and rational coefficients are written in binary number system. Theorem 1. Theory T is NP-hard and if is everywhere dense then the theory T is decidable by an algorithm which belongs to EXPTIME. The set of all rational numbers or the set of all binary rational numbers may be taken instead of . The quantifier-free negationless theory of the field of rational numbers may be regarded as a base of constructive mathematics. (shrink)

We provide a detailed history of the concepts of atomic number and isotopy before the discovery of protons and neutrons that draws attention to the role of evolving interplays of multiple aims and criteria in chemical and physical research. Focusing on research by Frederick Soddy and Ernest Rutherford, we show that, in the context of differentiating disciplinary projects, the adoption of a complex and shifting concept of elemental identity and the ordering role of the periodic table led to a (...) relatively coherent notion of atomic number. Subsequent attention to valency, still neglected in the secondary literature, and to nuclear charge led to a decoupling of the concepts of elemental identity and weight and allowed for a coherent concept of isotopy. This concept received motivation from empirical investigations on the decomposition series of radioelements and their unstable chemical identity. A new model of chemical order was the result of an ongoing collaboration between chemical and physical research projects with evolving aims and standards. After key concepts were considered resolved and their territories were clarified, chemistry and physics resumed autonomous projects, yet remained bound by newly accepted explanatory relations. It is an episode of scientific collaboration and partial integration without simple, wholesale gestalt switches or chemical revolutions. (shrink)

There is a need to increase the number of practicing medical doctors in South Africa. We examine the ethical implications of patients’ rights being affected in medical education in a South African context.The South African legal framework advocates public healthcare access. Yet, the State’s ethical obligations when it comes to guaranteeing public healthcare access, conflict with its utilitarian policy, that allows for medical education to help achieve the State’s public healthcare commitments, at the cost of eroding patients’ rights, and accepts (...) that certain actions are imperative, in line with Ubuntu, which is tenable yet nuanced. A patient treated by a licenced doctor today, benefits because other patients have allowed themselves to be used as hands-on learning material for medical students yesterday.Healthcare institutions need to take cognisance of the numbers of medical students that patients can reasonably be expected to endure. There is a need for the Health Professions Council of South Africa and medical schools to adopt guidelines on reasonable levels of medical student-patient interaction, and medical student-to-patient ratios in healthcare delivery. (shrink)

Let [Formula: see text] be an integer. We show that the set of real numbers that are Poisson generic in base [Formula: see text] is [Formula: see text]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base [Formula: see text] and not Poisson generic in base [Formula: see text] is complete for the class given by the differences between [Formula: see text] sets. We also show (...) that the effective versions of these results hold in the effective Borel hierarchy. (shrink)

The paper is aimed at the issues of oxidation state determination and limiting values. The possibility of existence of compounds containing an atom with the oxidation number beyond the current common values, i.e., below −IV and above +VIII are discussed. Three principal modes of preparation of compounds with the oxidation number exceeding VIII, electrochemical anodic oxidation, photoionization, and nuclear β-decay, are evaluated. Failure to prepare compounds containing an atom with the oxidation number below −IV is rationalized. The paper provides an (...) opinion on uncertainties in oxidation state determination in three kinds of compounds: binary compounds, nitrosyl complexes, and compounds containing mutually bonded atoms of the same element. The questions are discussed from the viewpoint of correlation of “man-made” quantities and objective, experimentally obtainable data. (shrink)

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are (...) not primarily treated abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)

Journal of Mathematical Logic, Ahead of Print. Let [math] be an integer. We show that the set of real numbers that are Poisson generic in base [math] is [math]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base [math] and not Poisson generic in base [math] is complete for the class given by the differences between [math] sets. We also show that the effective versions of (...) these results hold in the effective Borel hierarchy. (shrink)

In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics, on the possibility of quantitative psychology, and on the meaning of temperature measurement. Late nineteenth-century scrutinisers of the foundations of mathematics made little of Helmholtz’s essay. Yet it inspired two mathematicians with an eye on physics, and a few philosopher-physicists. The aim of the present paper is to situate Helmholtz’s contribution (...) in this complex array of nineteenth-century philosophies of number, quantity, and measurement.Author Keywords: Helmholtz; Measurement; Arithmetic; P. Du Bois-Reymond; H. and R. Grassmann; J. von Kries. (shrink)

A “mental” multiplication scheme is given for the super hypercomplex numbers, which extend the 16-element Dirac algebra to 32 elements by appending the complex octonions. This extends the 5-vectors of relativity to 9-vectors. The problems with nonassociativity, for the group structures and wave equation covariance, are explored.

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, (...) but currently, in the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

Despite the appeal of ‘the greatest good for the greatest number’ as an ethical ideal for businesses to pursue, applying this utilitarian principle in practice proves challenging. This is not least due to fundamental disagreements as to what constitutes the ‘greatest good.’ For example, the concept of ‘wellbeing’ now commonly proposed as a way of apprehending the greatest good is itself subject to widely varying interpretations. Drawing on an in-depth qualitative study of 64 managers in different sectors and country contexts, (...) we explore this variation through the lens of constructivist ethics, asking how and why managers systematically differ in their ethical meaning-making around wellbeing. Our theorizing advances constructivist ethics by relating these differences to developmental stages identified in constructivist psychology, finding that systematic variations in ethical meaning-making are shaped by differences in actors’ capacities to process complexity. Our analysis reveals that managers’ ethical meaning-making about wellbeing is subjective, socially constructed, dynamic, and evolutionary, progressing in stages that we differentiate with a novel concept of ‘subjective wellbeing complexity.’ We contribute to practice by discussing how managers’ ability to work with more complex conceptions of wellbeing can be purposefully enhanced through stage-by-stage capacity-building in the form of ‘vertical development.’. (shrink)

ABSTRACT This article explores the effects of scientific governance on personal liberty in interwar Britain through the work and life of German-Jewish demographer Robert René Kuczynski. Kuczynski arrived in Britain as a refugee in 1933 and, within the span of a few years, moved from being a researcher and reader at the London School of Economics to becoming demographic adviser to the Colonial Office. In the service of the British government, Kuczynski realized the first complete demographic survey of the British (...) Empire. Based on extensive primary research at the London School of Economics Archives and the Zentral- und Landesbibliothek Berlin, this article analyses the complex ways in which Kuczynski’s experience as a forced intellectual migrant interacted with – and often contradicted – his scientific work on population and the national and imperial policies that were enabled by it. Doing so, this article points to the inadequacy of merely evaluating personal freedom by means of law and power politics and asks about the hidden constraints in technocratic governance based on scientific knowledge. (shrink)

The convergence of the Neoplatonic/Neopythagorean approach with the Aristotelian organization of the sciences is one of the most interesting features that characterizes the two influential mathematical treatises on On Arithmetics and On Music by Severinus Boethius. Basing his reasoning on Nicomachus and Ptolemy, Boethius follows the philosophical tradition that had tried to reconcile Plato’s and Aristotle’s views. This attitude is examined in the present paper as regards Boethius’ response concerning the relation between numbers, ethics and aesthetics. His view emerges (...) as coming out of a rather complex construction, which assigns the ethical scope of mathematics in indicating to the human mind how to correct the ratios that realize the best relationship in movements of the soul and the body. More precisely, its ethical aim is to correct the specific form of movement of human beings, that is their actions, exemplified in the way in which mathematical ratios represent the forms of government and musical ratios evoke and heal psychophysical affections. More complex, on the other hand, is the relationship between mathematics and beauty. In clear antithesis to the position taken by Augustine on the beauty of the rhythmic patterns that better represent the beauty of unity, Boethius does not relate the mathematical ratios of the consonances to an esthetical judgment by making use of the category of beauty. For him, the physical world is totally immersed in changes and movements, and this cannot but impede things from expressing the stable unity, which is required for contemplating the beautiful. (shrink)

In this paper, we propose a method to design the pseudorandom number generator using three kinds of four-wing memristive hyperchaotic systems with different dimensions as multientropy sources. The principle of this method is to obtain pseudorandom numbers with good randomness by coupling XOR operation on the three kinds of FWMHSs with different dimensions. In order to prove its potential application in secure communication, the security of PRNG based on this scheme is analyzed from the perspective of cryptography. In addition, (...) PRNG has passed the NIST 800.22 and ENT test, which shows that PRNG has good statistical characteristics. Finally, an image encryption algorithm based on PRNG is adopted. In the encryption algorithm, the optimized Arnold matrix scrambling method and the diffusion processing based on XOR are used to obtain the final encrypted image. Through the evaluation of encryption performance, it is concluded that there is no direct relationship between the pristine image and encrypted image. The results show that the proposed image encryption scheme has good statistical output characteristics and security performance in line with cryptography. (shrink)

A psychologist offers a detailed study of the genetic underpinnings of human thought, looking at the small number of genes that contain the instructions for building the vastly complex human brain to determine how these genes work, common misconceptions about genes, and their implications for the future of genetic engineering. 30,000 first printing.

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. (...) class='Hi'>Numbers and arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

Quantitative information is one of the means used to interface science with policy. As a consequence, much effort is invested in producing quantitative information for policy and much criticism is directed toward the use of numbers in policy. In this paper, I analyze five approaches drawn from such criticisms and propose alternative uses of quantitative information for governance: valuation of ecosystem services, social multicriteria evaluation, quantification of uncertainty through the Numeral, Unit, Spread, Assessment, Pedigree approach, Quantitative Story-Telling, and the (...) heuristic use of statistics. The analysis shows the varied ways that numbers are conceptualized and how different conceptualizations matter for the science–policy interface. Alternative conceptualizations of numbers are used to challenge the model of science-speaking truth to power. Uncertainty, complexity, pluralism, malpractice, and values are mobilized to redefine the relations between science and policy. Alternative quantification may produce alternative facts, but reflexive approaches that use numbers to discuss the relevance of equity, positionality, and quality in science for policy may offer a remedy. (shrink)

This essay poses the question of how many rulers are envisaged in Plato’s Statesman. After pointing out that this is a crucial question for issues concerning non-ideal as well as ideal approaches to political rule, the essay focuses on three relevant aspects of rule in the Statesman: the notion of kingly rule, the limitations posed by human nature, and the importance of self-rule. It is shown how each of these dimensions of Plato’s discussion demonstrates the complexity of the question. Particular (...) attention is then given to features inherent to political rule: the need for subordinate functions and a distribution of offices, seen in light of the ends of political rule as helping citizens obtain their potential. It is argued that while the Statesman does not lead to any certain conclusion concerning the number of rulers, and some of its considerations conflict with each other, the text as a whole allows for a fairly broad basis of political rule. (shrink)

The conformity of mathematics and physics has so far been taken for granted. Philosophical explanations of that fundamental fact have never been satisfactory, mathematical explanations never had been attempted. In the following a fundamental theorem for the conformity of mathematics and physics will be demonstrated.Mathematics can be defined as the science of Number, physics as the science of Matter. The elementary constituents of mathematics are the prime numbers, those of matter the particles, particularly protons and electrons. The only essential (...) property of these elements as such is their existence, given within a given complex or space. Their fundamental relation is their distribution within their respective spaces. The theorem of the distribution of primes in the arithmetical series of numbers is the fundamental theorem of mathematics. The fundamental theorem of physics would be a theorem of the distribution of particles in geometrical space. If it were shown that both theorems were equal, the essential conformity of mathematics and physics would be mathematically demonstrated. (shrink)

We prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[n k ] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = (...) VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity. (shrink)

First language attrition in adulthood offers new insight on neuroplasticity and the role of language experience in shaping neurocognitive responses to language. Attriters are multilinguals for whom advancing L2 proficiency comes at the cost of the L1, as they experience a shift in exposure and dominance. To date, the neurocognitive mechanisms underlying L1 attrition are largely unexplored. Using event-related potentials, we examined L1-Italian grammatical processing in 24 attriters and 30 Italian native-controls. We assessed whether attriters differed from non-attriting native speakers (...) in their online detection and re-analysis/repair of number agreement violations, and whether differences in processing were modulated by L1-proficiency. To test both local and non-local agreement violations, we manipulated agreement between three inflected constituents and examined ERP responses on two of these. Our findings revealed group differences in amplitude, scalp distribution, and duration of LAN/N400 + P600 effects. We discuss these differences as reflecting influence of attriters’ L2-English, as well as shallower online sentence repair processes than in non-attriting native speakers. ERP responses were also predicted by L1-Italian proficiency scores, with smaller N400/P600 amplitudes in lower proficiency individuals. Proficiency only modulated P600 amplitude between 650 and 900 ms, whereas the late P600 depended on group membership and amount of L1 exposure within attriters. Our study is the first to show qualitative and quantitative differences in ERP responses in attriters compared to non-attriting native speakers. Our results also emphasize that proficiency predicts language processing profiles, even in native-speakers, and that the P600 should not be considered a monolithic component. (shrink)

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

There are nine kinds of number: cardinal (measure of class size), ordinal (corresponds to position), generalized ordinal (position in multidimensional discrete manifold), signed (relation between cardinals), rational (different kind of relation between cardinals), real (limit), complex (pair of reals), transfinite (size of reflexive class), and dimension (measure of complexity.

The Chemical Entities of Biological Interest (ChEBI) ontology is an important reference for applications dealing with chemical annotations and data mining. Modeling errors and inconsistencies in th...

In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbert-style formal systems such as in [17]. Thus a proof is a sequence offormulas satisfying certain conditions. We can view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length , depth and number of steps For a particular formal system and a given formula A we consider the shortest (...) length of a proof of A , the minimal depth of a proof of A and the minimal number of steps in a proof of A . The main results are the following: a bound on the depth in terms of the number of steps: Theorem 2.2, a bound on the depth in terms of the length: Theorem 2.3, a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: a bound on the number of steps in a cut-free proof, some speed-up results, bounds on the number of steps in proofs of Paris-Harrington sentences. Some papers related to our research are listed in the references. We were especially influenced by Parikh's paper [17] on the famous conjecture of Kreisel . Many important problems in this field remain open. We hope that our paper will contribute to progress in this area. It should be noted that some results of this paper can be equally well obtained using unification theory , by translating a complexity-of-proof problem into a unification problem. A unification algorithm solving the unification problem can then be constructed and the complexity of the unification algorithm analyzed. As pointed out by the referee this method can be used to solve the problem stated in Section 3 for some non-simple schematic systems. (shrink)

We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be (...) used to create polynomials which express the bits of Ω in the number of positive values they assume. (shrink)

In the 1830s and 1840s, the Hydrographic Office of the British Admiralty developed and oversaw one of the major state-run surveying projects of the nineteenth century. This involved a range of instruments whose circulation was increasingly regulated. Using extant museum collections and the correspondence of those involved, this article explores how such objects can be used to discuss both bureaucratic organization at a time of expanding government and the complex issues of sociability involved in hydrographic surveying. Surveying officers worked (...) in a context in which the propriety of property on public service was a pervasive question. Instruments might be given as gifts between officers, appropriated as recompense, absorbed as state property, and disputed between friends. The ownership, provision, and treatment of instruments in particular could be used to demonstrate an officer’s peculiar zeal or institutional neglect. To those outside the ship, what was understood as over-instrumentation became amusing spectacle. On board, their use was part of a deeply hierarchical order of work in regions of colonial and mercantile importance. In examining the relationships around these instruments of survey, the paper proposes a richer understanding of the material culture of hydrography in the early nineteenth century. (shrink)