Aristotle, Eleaticism, and Zeno’s Grains of Millet

This paper explores how Aristotle rejects some Eleatic tenets in general and some of Zeno’s views in particular that apparently threaten the Aristotelian “science of nature.” According to Zeno, it is impossible for a thing to traverse what is infinite or to come in contact with infinite things in a finite time. Aristotle takes the Zenonian view to be wrong by resorting to his distinction between potentiality and actuality and to his theory of mathematical proportions as applied to the motive power and the moved object (Ph. VII.5). He states that some minimal parts of certain magnitudes (i.e., continuous quantities) are perceived, but only in potentiality, not in actuality. This being so, Zeno’s view that a single grain of millet makes no sound on falling, but a thousand grains make a sound must be rejected. If Zeno’s paradoxes were true, there would be no motion, but if there is no motion, there is no nature, and hence, there cannot be a science of nature. What Aristotle noted in the millet seed paradox, I hold, is that it apparently casts doubt on his theory of mathematical proportions, i.e., the theory of proportions that holds between the moving power and the object moved, and the extent of the change and the time taken. This approach explains why Aristotle establishes an analogy between the millet seed paradox, on the one hand, and the argument of the stone being worn away by the drop of water (Ph. 253b15–16) and the hauled ship, on the other.

Logic, geometry and visualisation of the body in Acquapendente’s rediscovered Methodus anatomica (1579)

The article provides the first description and analysis of the recently rediscovered manuscript titled Methodus anatomica by Girolamo Fabrici da Acquapendente (1533–1619). Acquapendente was one of the most important anatomists in late sixteenth-century Europe and played an instrumental role as Harvey’s teacher in Padua towards the latter’s discovery of the circulation of the blood. The manuscript provides first-hand testimony as to how anatomy was administered in Padua in the post-Vesalian era and sheds light on a number of otherwise unknown aspects of the development of the anatomical method. Chiefly among these is the attention devoted by Acquapendente to historia, as a way to order sensory data in a consistent way, which draws widely from the geometrical method and from the contemporary debate on the discretisation of continuous quantities.

Continuous versus discrete quantity discrimination in dune snail (Mollusca: Gastropoda) seeking thermal refuges

The ability of invertebrates to discriminate quantities is poorly studied, and it is unknown whether other phyla possess the same richness and sophistication of quantification mechanisms observed in vertebrates. The dune snail, Theba pisana, occupies a harsh habitat characterised by sparse vegetation and diurnal soil temperatures well above the thermal tolerance of this species. To survive, a snail must locate and climb one of the rare tall herbs each dawn and spend the daytime hours in an elevated refuge position. Based on their ecology, we predicted that dune snails would prefer larger to smaller groups of refuges. We simulated shelter choice under controlled laboratory conditions. Snails’ acuity in discriminating quantity of shelters was comparable to that of mammals and birds, reaching the 4 versus 5 item discrimination, suggesting that natural selection could drive the evolution of advanced cognitive abilities even in small-brained animals if these functions have a high survival value. In a subsequent series of experiments, we investigated whether snails used numerical information or based their decisions upon continuous quantities, such as cumulative surface, density or convex hull, which co-varies with number. Though our results tend to underplay the role of these continuous cues, behavioural data alone are insufficient to determine if dune snails were using numerical information, leaving open the question of whether gastropod molluscans possess elementary abilities for numerical processing.

On the Sizing of the DC-Link Capacitor to Increase the Power Transfer in a Series-Series Inductive Resonant Wireless Charging Station

Wireless inductive-coupled power transfer is a very appealing technique for the battery recharge of autonomous devices like surveillance drones. The charger design mainly focuses on lightness and fast-charging to improve the drone mission times and reduce the no-flight gaps. The charger secondary circuit mounted on the drone generally consists of a full-bridge rectifier and a second-order filter. The filter cut-off frequency is usually chosen to make the rectifier output voltage constant and so that the battery is charged with continuous quantities. Previous works showed that an increase in power transfer is achieved, if compared to the traditional case, when the second-order filter resonant frequency is close to the double of the wireless charger excitation and the filter works in resonance. This work demonstrates that the condition of resonance is necessary but not sufficient to achieve the power increment. The bridge rectifier diodes must work in discontinuous-mode to improve the power transfer. The paper also investigates the dependence of the power transfer increase on the wireless excitation frequency. It is found the minimum frequency value below which the power transfer gain is not possible. This frequency transition point is calculated, and it is shown that the gain in power transfer is obtained for any battery when its equivalent circuit parameters are known. LTSpice simulations demonstrate that the transferred power can be incremented of around 30%, if compared to the case in which the rectifier works in continuous mode. This achievement is obtained by following the design recommendations proposed at the end of the paper, which trade off the gain in power transfer and the amplitude of the oscillating components of the wireless charger output.

The approximate number system represents magnitude and precision

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities (e.g., numerosities, durations, rates, distances, directions, and probabilities), but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

Desenvolvimento do Pensamento Teórico de Professores dos Anos Iniciais sobre FraçõesDevelopment of the theoretical thinking of early years teachers about fractions

ResumoO artigo analisa o desenvolvimento de aspectos do pensamento teórico do professor sobre frações, em particular: mediação de grandezas contínuas e a equivalência de frações. Apoiada teoricamente na perspectiva histórico-cultural e na Teoria da Atividade, a pesquisa adotou a Atividade Orientadora de Ensino (AOE) como referência para a organização de ações de uma formação continuada de professores. A análise dos dados produzidos junto aos professores revela a superação da ideia de fração como a quantificação discreta de partes já dadas de um inteiro, a criação de subunidade como estratégia para a quantificação de grandeza contínua e a apropriação do sentido de comparação de frações por meio de frações equivalentes. Concluímos que tais elementos revelam aspectos da superação do pensamento empírico pelo pensamento teórico.Palavras-chave: Teoria Histórico-Cultural; Fração; Formação de Professores. AbstractThe article analyzes the development of aspects of the teacher's theoretical thinking about fractions, in particular: the mediation of continuous quantities and the equivalence of fractions. Supported theoretically by the historical-cultural perspective and the Theory of the Activity, the research adopted the Teaching Guiding Activity (AOE) as a reference for the organization of continuous teacher education. The analysis of the data produced with teachers reveals the overcoming of the idea of the fraction as the discrete quantification of parts already given of a whole, the creation of a subunit as a strategy for the quantification of continuous magnitude and the appropriation of the sense of comparing fractions by means of equivalent fractions. We conclude that such elements reveal aspects of the overcoming of empirical thinking by theoretical thinking.Keywords: Cultural-Historical-Theory; Fraction; Teaching Guiding Activity. ResumenEl artículo analiza el desarrollo de aspectos del pensamiento teórico del profesor sobre fracciones, en particular: mediación de avances continuos y equivalencia de fracciones. Teóricamente apoyado en la perspectiva histórico-cultural y en la Teoría de la Actividad, una investigación adoptó la Actividad de Orientación Docente (AOE) como referencia para organizar acciones para la formación continua del profesorado. El análisis de los datos elegidos con los maestros revela una superación de la idea de fracción como una cuantificación de partes ya otorgadas un número entero, una creación de subunidad como estrategia para la cuantificación de magnitud continua y la apropiación del sentido de comparar fracciones por medio de fracciones equivalentes. Concluya que estos elementos revelan aspectos de superar el pensamiento empírico mediante el pensamiento teórico.Palabras clave: Teoría Histórico-Cultural; Fracción; Formación del profesorado

How does a fraction get its name?

Philosophical and cultural perspectives shape how a fraction is named and defined. In turn, these perspectives have consequences for learners' conceptualization of fractions. We examine historical foundations of two perspectives of what are fractions—partitioning and measuring—and how these views influence fraction knowledge. For the dominant perspective, partitioning, we indicate how its approach to what is a fraction that discretizes objects and its well-meaning visual correlates cause learners a host of perceptual difficulties. Based on the human cultural and social practice of measuring continuous quantities, we then offer an alternative understanding of what is a fraction and illustrate the promise of this view for fraction knowledge. We introduce pedagogical tools, Cuisenaire rods, and illustrate how they can be used to implement a measuring perspective to comprehending properties and a definition of fractions. We end by sketching how to initiate a measuring perspective in a mathematics classroom.Keywors: Fractions; Gattegno; Measuring; Partitioning; Unit fractions. Como uma fração recebe seu nome?Resumo: Perspectivas filosóficas e culturais moldam como uma fração é nomeada e definida. Por sua vez, essas perspectivas têm consequências para a conceitualização de frações dos estudantes. Examinamos os fundamentos históricos de duas perspectivas do que são frações—particionamento e medição—e como essas visões influenciam o conhecimento das frações. Para a perspectiva dominante, partição, indicamos como sua abordagem ao que é uma fração que discretiza objetos e seu correlato visual bem-intencionado causa aos alunos uma série de dificuldades perceptivas. Com base na prática cultural e social humana de medir quantidades contínuas, oferecemos um entendimento alternativo do que é uma fração e ilustramos a promessa dessa visão para o conhecimento da fração. Introduzimos ferramentas pedagógicas, varas Cuisenaire e ilustramos como elas podem ser usadas para implementar uma perspectiva de medição para compreender propriedades e uma definição de frações. Terminamos esboçando como iniciar uma perspectiva de medição em uma sala de aula de matemática.Palavras-chave: Frações; Gattegno; Medição; Partição; Frações unitárias.

Extending Classical Planning with State Constraints: Heuristics and Search for Optimal Planning

We present a principled way of extending a classical AI planning formalism with systems of state constraints, which relate - sometimes determine - the values of variables in each state traversed by the plan. This extension occupies an attractive middle ground between expressivity and complexity. It enables modelling a new range of problems, as well as formulating more efficient models of classical planning problems. An example of the former is planning-based control of networked physical systems - power networks, for example - in which a local, discrete control action can have global effects on continuous quantities, such as altering flows across the entire network. At the same time, our extension remains decidable as long as the satisfiability of sets of state constraints is decidable, including in the presence of numeric state variables, and we demonstrate that effective techniques for cost-optimal planning known in the classical setting - in particular, relaxation-based admissible heuristics - can be adapted to the extended formalism. In this paper, we apply our approach to constraints in the form of linear or non-linear equations over numeric state variables, but the approach is independent of the type of state constraints, as long as there exists a procedure that decides their consistency. The planner and the constraint solver interact through a well-defined, narrow interface, in which the solver requires no specialisation to the planning context.

Continuum dynamics

The theory and application of a variety of methods to solve partial differential equations are introduced in this chapter. These methods rely on representing continuous quantities with discrete approximations. The resulting finite difference equations are solved using algorithms that stress different traits, such as stability or accuracy. The Crank-Nicolson method is described and extended to multidimensional partial differential equations via the technique of operator splitting. An application to the time-dependent Schrödinger equation, via scattering from a barrier, follows. Methods for solving boundary value problems are explored next. One of these is the ubiquitous fast Fourier transform which permits the accurate solution of problems with simple boundary conditions. Lastly, the finite element method that is central to modern engineering is developed. Methods for generating finite element meshes and estimating errors are also discussed.

Differential form representation of stochastic electromagnetic fields

In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.