We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with (...) further structure. (shrink)

This article presents the Arabic text of al-Q's short geometric treatise, The Book of the centres of the tangent circles on lines, by the method of analysis, with translation and a mathematical commentary. In this treatise al-Q solves by analysis an ordered set of eight problems where the goal is to locate, on a given line, the centre of a circle which is tangent to two given elements, which may be points, straight lines or circles. For example, in (...) the first problem he wanted to find the centre of a circle which contains two given points. In fact, al-Q establishes the locus of the solutions, and this locus can be a straight line or a conic. The treatise is reminiscent of one of the purposes of al-Q's work on the perfect compass, namely to assign to the conics the status of regular lines. (shrink)

Objective: Tangent Space Mapping using the geometric structure of the covariance matrices is an effective method to recognize multiclass motor imagery. Compared with the traditional CSP method, the Riemann geometric method based on TSM takes into account the nonlinear information contained in the covariance matrix, and can extract more abundant and effective features. Moreover, the method is an unsupervised operation, which can reduce the time of feature extraction. However, EEG features induced by MI mental activities of different subjects are (...) not the same, so selection of subject-specific discriminative EEG frequency components play a vital role in the recognition of multiclass MI. In order to solve the problem, a discriminative and multi-scale filter bank tangent space mapping algorithm is proposed in this article to design the subject-specific Filter Bank so as to effectively recognize multiclass MI tasks. Methods: On the 4-class BCI competition IV-2a dataset, first, a non-parametric method of multivariate analysis of variance based on the sum of squared distances is used to select discriminative frequency bands for a subject; next, a multi-scale FB is generated according to the range of these frequency bands, and then decompose multi-channel EEG of the subject into multiple sub-bands combined with several time windows. Then TSM algorithm is used to estimate Riemannian tangent space features in each sub-band and finally a liner Support Vector Machines is used for classification. Main Results: The analysis results show that the proposed discriminative FB enhances the multi-scale TSM algorithm, improves the classification accuracy and reduces the execution time during training and testing. On the 4-class BCI competition IV-2a dataset, the average session to session classification accuracy of nine subjects reached 77.33 ± 12.3%. When the training time and the test time are similar, the average classification accuracy is 2.56% higher than the latest TSM method based on multi-scale filter bank analysis technology. When the classification accuracy is similar, the training speed is increased by more than three times, and the test speed is increased two times more. Compared with Supervised Fisher Geodesic Minimum Distance to the Mean, another new variant based on Riemann geometry classifier, the average accuracy is 3.36% higher, we also compared with the latest Deep Learning method, and the average accuracy of 10-fold cross validation improved by 2.58%. Conclusion: Research shows that the proposed DMFBTSM algorithm can improve the classification accuracy of MI tasks. Significance: Compared with the MFBTSM algorithm, the algorithm proposed in this article is expected to select frequency bands with good separability for specific subject to improve the classification accuracy of multiclass MI tasks and reduce the feature dimension to reduce training time and testing time. (shrink)

L’œuvre mathématique de Leibniz a ceci d’intéressant qu’au travers des innombrables manuscrits de travail dont nous disposons dans ses archives à Hanovre, le philosophe nous a confié le matériel nécessaire pour rétablir ses divers cheminements de recherche ainsi que ses méthodes de découvertes à l’origine de ses créations mathématiques. L’exemple des exposants que nous allons traiter permet d’éclairer utilement la façon dont Leibniz apprend les mathématiques et change progressivement de posture et de démarche. Ainsi, dans sa première année parisienne, la (...) primauté est-elle donnée aux relations tabulaires, plutôt qu’aux formes et aux similitudes. Ces enjeux épistémologiques expliquent la manière dont Leibniz généralise partiellement la notion d’exposant. (shrink)

Recent interest in maximal proper acceleration as a possible principle generalizing the theory of relativity can draw on the differential geometry of tangent bundles, pioneered by K. Yano, E. T. Davies, and S. Ishihara. The differential equations of geodesics of the spacetime tangent bundle are reduced and investigated in the special case of a Riemannian spacetime base manifold. Simple relations are described between the natural lift of ordinary spacetime geodesics and geodesics in the spacetime tangent bundle.

Este trabajo muestra una concurrencia patente en la resonancia de las categorías ‘industria cultural’ [Kulturindustrie] –empleada por Theodor Adorno para indicar ciertas zonas de la dialéctica subjetiva– e ‘industria de recreo’ [Vergnügungsindustrie] –referida por Walter Benjamin en relación a la dialéctica objetiva–. Me detengo en esta poco explorada concomitancia para determinar por qué Adorno evidencia una dialéctica de la subjetividad administrada en la renuncia sacrificial y Benjamin elabora una dialéctica de la objetividad fetichizada signada por su carácter de fantasmagoría cadavérica. (...) Así, señalo una dimensión de la incardinación entre la veta expansiva de la racionalidad moderna y la instrumentalización de sujetos y objetos. (shrink)

Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

This is the first volume compiling English translations of Leibniz's journal articles on natural philosophy, presenting a selection of 26 articles, only three of which have appeared before in English translation. It also includes in full Leibniz's public controversies with De Catelan, Papin, and Hartsoeker. The articles include work in optics, on the fracture strength of materials, and on motion in a resisting medium, and Leibniz's pioneering applications of his calculus to these issues by construing them as mini-max and inverse (...) tangent problems. Other topics covered by the articles include: criticisms of the Cartesian estimate of motive force and Leibniz's proposal of a different way of estimating force to replace it; a proposed theory of celestial motions and gravitation, and derivation of the inverse square law; challenge problems concerning the isochronous curve and the catenary; a sample of work on gaming theory; and Leibniz's critique of atomism. (shrink)

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

The standard view is that the Lagrangian and Hamiltonian formulations of classical mechanics are theoretically equivalent. Jill North, however, argues that they are not. In particular, she argues that the state-space of Hamiltonian mechanics has less structure than the state-space of Lagrangian mechanics. I will isolate two arguments that North puts forward for this conclusion and argue that neither yet succeeds. 1 Introduction2 Hamiltonian State-space Has less Structure than Lagrangian State-space2.1 Lagrangian state-space is metrical2.2 Hamiltonian state-space is symplectic2.3 Metric > (...) symplectic3 Hamiltonian State-space Does Not Have Less Structure than Lagrangian State-space3.1 Lagrangian state-space has less than metric structure3.1.1 A potential worry3.1.2 General Lagrangians3.2 Hamiltonian state-space has more than symplectic structure3.2.1 A dual structure on the Hamiltonian state-space3.2.2 Simple Hamiltonians3.3 Comparing Lagrangian and Hamiltonian structures3.3.1 Counting mathematical structure3.3.2 Symplectic and metric structure are incomparable4 An Alternative Argument for LS4.1 The argument for P3*4.2 Symplectic manifold vs. tangent bundle structure4.3 Trying to patch up the argument for P3*5 Interpreting Mathematical Structure5.1 State-space realism5.2 Model isomorphism and theoretical equivalence6 Conclusion. (shrink)

Love has been a topic of interest to philosophy since at least the time of Plato’s Symposium, but with a few notable exceptions, it was unduly neglected in the twentieth century, at least by writers in the analytic tradition that predominates in the English-speaking world. However, in the past quarter century, writing on the topic has exploded. In this volume, we touch on most of the currently hot debates and also introduce some fascinating tangents. The main threads of discussion reflected (...) in this volume are as follows: the relationship between love and morality ; whether love is rational, subject to reasons for or against it, or a force that is not under our intentional control; and whether love affects the way we perceive the world or the way we value things in the world. More singular topics include: whether love would be affected by disputes in the literature on free will; whether we could be mistaken about being in love; whether our pets are capable of loving us back; whether a relationship of the kind shown in the movie her between a human and an artificial intelligence could be either loving or ethical; and whether the difference between patriotism and nationalism hinges on how each instances a different kind of love. Along the way we will see analyses of the work of philosophical greats like Immanuel Kant as well as the work of more contemporary writers, in particular Iris Murdoch, and philosophers actively engaged in the current revival, notably Harry Frankfurt, J. David Velleman, and Niko Kolodny. (shrink)

In this article, we assess the heuristic power of Greek geometrical analysis by trying to reconstruct some analyses of extant propositions of which only the demonstration is found in the text. We have reconstructed the analysis of the trisection of an angle, the property of the tangent to the parabola, to the hyperbola/ellipse, and to the spiral line. In all of these cases, the results and the demonstrations can be found by the analysis alone, without arguments by analogy with (...) other figures or kinematic considerations. We have thus concluded that analysis is a much more powerful tool than is usually assumed to be. We have also examined a part of Apollonius’ Cutting off of a ratio and confirmed that the number of possible cases to be distinguished in analysis tended to increase because the geometric objects and the magnitudes related to them were always considered in the geometric arrangement in which they appear. We suggest that the absence of an algebraic approach that would have detached the magnitudes from their geometrical context may have been one of the major obstacles to further development of Greek geometry. (shrink)

Abstract.As an undergraduate at Cambridge, Newton entered into his ‘Waste Book’ an assumption that we have named the Equivalence Assumption (The Younger): ‘‘ If a body move progressively in some crooked line [about a center of motion]..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent.’’ In this assumption, Newton somewhat imprecisely describes (...) two mathematical models, a ‘‘polygonal limit model’’ and a ‘‘tangent deflected model,’’ for ‘‘one-body motion,’’ that is, for the motion of a ‘‘body in orbit about a fixed center,’’ and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newton’s work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibniz’s ‘‘polygonal model’’ of one-body motion; then to repair Newton’s argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in ‘‘De motu’’ and the Principia. (shrink)

Abstract: The Pull Bayesian Significance Test (FBST) for precise hy- potheses is applied to a Multivariate Normal Structure (MNS) model. In the FBST we compute the evidence against the precise hypothesis. This evi- dence is the probability of the Highest Relative Surprise Set (HRSS) tangent to the sub-manifold (of the parameter space) that defines the null hypothesis. The MNS model we present appears when testing equivalence conditions for genetic expression measurements, using micro-array technology.

Vagueness does not necessarily come in with vague predicates, nor need it be expressed by them , but undoubtedly 'vague predicates' are traditionally in the focus of main stream discussions of vagueness. In her current modal logic presentation and discussion of the Sorites paradox Susanne Bobzien[1] lists among the properties of a Sorites series a rather weak modal tolerance principle governing the 'grey zone' containing the borderline cases of the Sorites series, which later proves crucial for her solution of the (...) Sorites paradox by use of epistemic interpreted modal operators in 1st order modal logic. We suggest (for different research interest) instead a non-modal description of the switch in the grey zone (respecting tolerance), by resort to similarity sequences, thus getting tangent to two other areas of research in the field. Let's say -any way- the Sorites paradox vanishes, the Sorites series does not. (shrink)

The first proposition of the Principia records two fundamental properties of an orbital motion: the Fixed Plane Property (that the orbit lies in a fixed plane) and the Area Property (that the radius sweeps out equal areas in equal times). Taking at the start the traditional view, that by an orbital motion Newton means a centripetal motion – this is a motion ``continually deflected from the tangent toward a fixed center'' – we describe two serious flaws in the Principia's (...) argument for Proposition 1, an argument based on a polygonal impulse approximation. First, the persuasiveness of the argument depends crucially on the validity of the Impulse Assumption: that every centripetal motion can be represented as a limit of polygonal impulse motions. Yet Newton tacitly takes the Impulse Assumption for granted. The resulting gap in the argument for Proposition 1 is serious, for only a nontrivial analysis, involving the careful estimation of accumulating local errors, verifies the Impulse Assumption. Second, Newton's polygonal approximation scheme has an inherent and ultimately fatal disability: it does not establish nor can it be adapted to establish the Fixed Plane Property. Taking then a different view of what Newton means by an orbital motion – namely that an orbital motion is by definition a limit of polygonal impulse motions – we show in this case that polygonal approximation can be used to establish both the fixed plane and area properties without too much trouble, but that Newton's own argument still has flaws. Moreover, a crucial question, haunted by error accumulation and planarity problems, now arises: How plentiful are these differently defined orbital motions? Returning to the traditional view, that Newton's orbital motions are by definition centripetal motions, we go on to give three proofs of the Area Property which Newton ``could have given'' – two using polygonal approximation and a third using curvature – as well as a proof of the Fixed Plane Property which he ``almost could have given.''. (shrink)

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry based on certain algebraic objects, which regiments a mode of reasoning heuristically used by geometricists and physicists. I argue that SIG has the following utilities. It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. It generalizes a standard implementation of spacetime algebraicism called Einstein algebras. It solves the long-standing (...) problem of interpreting smooth infinitesimal analysis realistically, an alternative foundation of spacetime theories to real analysis, 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations. Against this, I argue that SIG is such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. (shrink)

Teaching through discussion is perhaps the most fundamental pedagogical technique of philosophy. When done well, discussion can promote long-term, durable learning. It supports an active-learning classroom, sparks higher-order cognition, and helps students construct their own understanding of the material. Yet it is often not done well: it is easily dominated by a few students; it can waste time on tangents and minutiae; it can fail to motivate students to prepare adequately; and it is not easily scalable. In this paper we (...) present a technique, the random discussion leader (RDL), which addresses all of these shortcomings of traditional discussion. Qualitative and quantitative evidence from our quasi-experimental study (n = 279) suggests that RDL leads to better preparation, better engagement with the material, more equitable participation, and higher quality discussions. Additionally, we provide a toolkit for instructors interested in trying RDL, showing how the technique can be adapted to various course settings and materials. (shrink)

The existence of a definite tangent space structure (metric with Lorentzian signature) in the general theory of relativity is the consequence of a fundamental assumption concerning the local validity of special relativity. There is then at the heart of Einstein's theory of gravity an absolute element which depends essentially on a common feature of all the non-gravitational interactions in the world, and which has nothing to do with space-time curvature. Tentative implications of this point for the significance of the (...) vacuum solutions in general relativity, and for the issue of quantising gravity, are briefly examined. (shrink)

Early in his mathematical career Leibniz discovered some important methods and results but had to recognize that his findings had been anticipated by other mathematicians such as Pierre de Fermat, James Gregory, Isaac Newton, François Regnauld, John Wallis, etc. This paper investigates the cases of Isaac Barrow and Pietro Mengoli who, earlier than Leibniz, had been familiar with the characteristic triangle, transmutations methods, the inverse connection between determining tangents and areas of curves or the sums of the reciprocal figurate numbers, (...) and the harmonic triangle. To what extent was Leibniz aware of the results and publications of his predecessors? How did he assess their methods and results? Why did Leibniz never acknowledge any influence of these two mathematicians on his own studies? After publication of Leibniz’s manuscripts concerning the prehistory and early history of the calculus in the Academy Edition these questions can be investigated on the solid foundation of original texts. (shrink)

O texto que aqui se apresenta tem por objetivo principal o desvelamento das possíveis relações entre a noção de ser-para-a-morte desenvolvida pelo filósofo alemão Martin Heidegger (1889-1976) em sua analítica existencial do Dasein desenvolvida em sua obra precípua, Ser e tempo, em vias de se empreender uma Ontologia Fundamental, e seus fenômenos tangentes, angústia e processo apropriativo, e os fazeres da psicoterapia.

Entre 1676 y 1680 Leibniz intentó conseguir un puesto remunerado en la Académie des sciencesque quedó vacante tras el fallecimiento de Roberval en 1675. Al mismo tiempo, Leibniz ofrece en cartas a Huygens varios métodos, recetas e invenciones que podrían ser de utilidad para la academia parisina: entre ellos encontramos el analysis situs, una discusión sobre el método de Becher para encontrar oro, la receta del fósforo, el método inverso de tangentes, la cuadratura aritmética y su propia red epistolar. La (...) hipótesis de este trabajo defiende que estas discusiones fueron presentadas a Huygens con la finalidad de conseguir el puesto remunerado en la Académie, lo cual no ha sido lo suficientemente tenido en cuenta en la literatura científica. (shrink)

Giovanni Bianchini’s fifteenth-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy’s Almagest. The completely novel structure of Bianchini’s tables, here and in his Tabulae magistrales, was (...) taken over by Regiomontanus in the latter’s Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates. (shrink)

Este escrito traz uma abordagem sobre a visão de alguns estudantes do ensino fundamental II e do ensino médio de escolas públicas do interior da Bahia, sobre como pensam as suas atuais situações educacionais em meio à pandemia causada pela COVID-19, motivo o qual as aulas presencias foram totalmente suspensas. Sobre o prisma das (im)possibilidades do acesso às TICs pelos estudantes, observamos o possível alargamento das suas autonomias educacionais, bem como impulsionamento de suas posturas critica reflexivas e, a partir destas, (...) a busca pelo empreendimento de suas próprias aprendizagens escolares. Buscamos externar as dificuldades de aprendizado em tempos pandêmicos; as exclusões digitais, cujas discrepâncias tornaram-se mais evidentes mediante a crise ocasionada pelo coronavírus. A partir de abordagem qualitativa e atividade de escuta dos sujeitos colaboradores em seus lugares e contextos, utilizando o instrumento entrevistas on line, buscamos dialogar com alguns autores como (Lemos, 2015); (Nicolaci–da-Costa, 2009); (Lévy 1999); (Silva, 2011); (Freire, 2006); (Amiél, 2018) e com os estudantes, jovens e adolescentes colaboradores buscando traçar uma imagem da realidade tangente a partir da visão destes sujeitos. Palavras -chave: Escola pública. Pandemia. Estudantes. Exclusão digital. Autonomia. (shrink)

Bianchini called Tabulae magistrales a set of eight tables he compiled to solve problems in spherical astronomy. This set, which is the object of this paper, consists of auxiliary and trigonometric functions, including the sine and the tangent functions, for radii 10,000 and 60,000, and seems to be the first set of tables in Latin specifically devoted to mathematical tools for computational astronomy. Bianchini presented some of his tables in decimal form, which meant that for the first time one (...) of the oldest astronomical tradition, the sexagesimal base, was abandoned. (shrink)

Em Androides sonham com ovelhas elétricas?, Philip Dick recria a visão divina dos infinitos mundo possíveis a partir da perspectiva das criaturas, isto é, a partir da visão dos mundos extintos que lhe aparecem no momento de hesitação garantido pelo livre arbítrio. Minha hipótese é que a passagem dos mundos possíveis no intelecto divino à forma que assumem na indiferença da vontade das criaturas tem uma dimensão geométrica. Pensar a curva do mundo atual a partir da série dos mundos menos (...) perfeitos do Deus leibniziano é pensá-la a partir de suas infinitas retas tangentes, a partir, portanto, de suas distopias, que não são uma contradição em relação a esse mundo, mas tocam-no em algum ponto, são seu máximo desvio a partir de uma mínima diferença. (shrink)

For a standard Finsler F on a manifold M, the domain is the whole tangent bundle T M and the fundamental tensor g is positive-definite. However, in many cases, these two conditions hold in a relaxed form only, namely one has either a psuedo-Finsler metric or a conic Finsler metric. Our aim is twofold. First, we want to give an account of quite a few subtleties that appear under such generalizations, say, for conic pseudo-finsler metrics. Second, we aim to (...) provide some criteria that determine when a pseudo-Finsler metric F obtained as a general homogeneous combination of Finsler metrics and one-forms is again a Finsler-metric -- or, more precisely, that the conic domain on which g remains positive-definite. Such a combination generalizes the known -metrics in different directions. Remarkably, classical examples of Finsler metrics are reobtained and extended, with explicit computations of their fundamental tensors. (shrink)

Let AC be a ray coming from the medium DC into the medium CE, and let the density of the former to the latter be as d to e. It is asked, how should the ray ACB be directed so that it is the easiest path of all, or that (AC x d) + (CB x e) is a minimum. Let DC = l, and EC = m. It is given also that FG = f, and let AD = FC (...) = x, CG = EB = f – x. Therefore, AC = √(l2 + x2) and CB = √(m2 + f2 +x2 – 2fx). 2 It will then be the case that d√(l2 + x2) + e√(m2 + f2 + x2 – 2fx is equal to a minimum. Therefore, through my method of tangents it is the case that: (2dx/(√(l2 +x2)(AC))) + ((2ex -2ef)/ (√(m2 +f2 +x2 +2fx)(BC))) = 0. That is, it will be the case that AC/BC = dx/e(f-x). Now if we suppose that AC and BC are equal, it will be the case that f – x is to x, as d to e. Therefore, if a circle, with its center at C, is described by the ray CA or CB, AD or “x,” – the sine of the angle of incidence – will be to BE or “f – x,” – the sine of the angle of refraction – as e, the density of the medium of refraction, will be to d, the density of the medium of incidence, that is, the sines of the angles will be in reciprocal relation to the mediums or densities. (shrink)

Background and ObjectiveAbsolute angle represents the inclination of a body segment relative to a fixed reference in space. This work compares the absolute and relative angles for exploring biomechanical gait constraints.MethodsGait patterns of different neuromotor conditions were analyzed using 3D gait analysis: normal gait, Cerebral Palsy, Charcot Marie Tooth and Duchenne Muscular Dystrophy, representing central and peripheral nervous system and muscular disorders, respectively. Forty-two children underwent gait analysis: 10 children affected by CP, 10 children by CMT, 10 children by DMD (...) and 12 healthy children. The kinematic and kinetic parameters were collected to describe the biomechanical pattern of participants’ lower limbs. The absolute angles of thigh, leg and foot were calculated using the trigonometric relationship of the tangent. For each absolute series, the mean, range, maximum, minimum and initial contact were calculated. Kinematic and kinetic gait data were studied, and the results were compared with the literature.ResultsStatistical analysis of the absolute angles showed how, at the local level, the single segments behave differently depending on the pathology. However, if the lower limb is studied globally, a biomechanical constraint emerges.ConclusionEach segment compensates separately for the disease deficit so as to maintain a global biomechanical invariance. Using a model of inter-joint co-variation could improve the interpretation of the clinical gait pattern. (shrink)

RésuméLe problème de la détermination de la Qibla est l'une des questions cruciales qui se posent à la culture scientifique de l'Islam médiéval; le résoudre correctement nécessite tant des théories mathématiques que des observations. Les mathématiques relèvent de deux chapitres: la trigonométrie plane et la trigonométrie sphérique. L'observation et les instruments d'observation sont indispensables à la détermination des coordonnées géographiques de La Mecque et du lieu donné; ces coordonnées sont en effet les données que l'on entre dans les formules donnant (...) la Qibla. Dans son Almageste, Abū al-Wafāʾ résout brillamment ce problème. Son travail porte tant sur les mathématiques que sur l'observation; il obtient des solutions modernes, adéquates et faciles. En trigonométrie plane, il donne de nouvelles définitions des fonctions trigonométriques, démontre les formules des sinus, donne une approximation de sin 1° grâce à laquelle il construit des tables de sinus et de tangentes d'une grande précision. En trigonométrie sphérique, il démontre quatre nouveaux théorèmes dont la règle des tangentes ; cette règle lui permet de trouver, comme nous le montrerons, des solutions simples au problème de la détermination de la Qibla. En ce qui concerne les observations, il décrit trois instruments utilisés par lui plusieurs années de suite à Bagdad. Cet article, nourri de nombreux textes arabes originaux traduits et commentés, donne une description détaillée, technique et analytique, des méthodes mathématiques d'Abū al-Wafāʾ pour la détermination de la Qibla. (shrink)

Illustrations: 31 Figures Description: The present work is a masterful compilation which, from its nature and scope, and from the method followed in its preparation, exhibits that it is not merely a dictionary, but it is in fact more than that. Arranged alphabetically, this dictionary has been conceived on a large scale and written by a deft hand. The work is divided into two volumes. Volume I deals with India and the Middle East and contains 2250 titles, and volume II (...) deals with China and Japan and has 1500 titles, and together they cover the vast field of Oriental philosophy, encompassing all human experiences and disciplines. Captured in the pages of this monumental work are the lives of the prophets, social and religious reformers, Asian philosophies and spiritual disciplines of the living faiths of the east. The dictionary is destined to be welcomed as an exhaustive work on comparative religion also, inasmuch as it covers the whole ground of Oriental philosophy and its tangents. The entire volume is a combination of scholarly knowledge and spiritual awareness; their importance therefore is both to scholars and seekers of truth. There is perhaps no other work on record that accomplishes both ends-the propounding of the custom of the peoples of the Orient and an insight into their spiritual ideals both ancient and modern. All in all, the Dictionary of Oriental Philosophy may be judges to have been written by a hand of master dialectician with perception and exactitude. (shrink)

The mechanical aspect of momentum, basically its role as a tangent vector of the trajectory of the particle, is related to properties of the momentum found in the contexts of Hamilton's optico-mechanical analogy, de Broglie's matter waves, and quantum mechanics. These properties are treated in a systematic way by considering an approximation of the particle mechanical action of the particle by a step function. A special method of discretizing partial differential equations is shown to be required. Using this method, (...) a discrete dynamics is developed. It is shown that particle dynamics can be regarded as the limit case of the discrete dynamics as the step functions tend to the continuous ones. The equation of motion of a free particle in an arbitrary reference system is deduced in two ways: (i) in continuous dynamics by making use of the invariance of action within changes of reference systems, and (ii) by taking the mentioned limit in discrete dynamics of an equation which expresses that the mechanical and wave-theoretical aspects of the momentum are interrelated in specific way. (shrink)

It is shown that relativistic spacetimes can be viewed as Finslerian spaces endowed with a positive definite distance (ω0, mod ωi) rather than as pariah, pseudo-Riemannian spaces. Since the pursuit of better implementations of “Euclidicity in the small” advocates absolute parallelism, teleparallel nonlinear Euclidean (i.e., Finslerian) connections are scrutinized. The fact that (ωμ, ω0 i) is the set of horizontal fundamental 1-forms in the Finslerian fibration implies that it can be used in principle for obtainingcompatible new structures. If the connection (...) is teleparallel, a Kaluza-Klein space (KKS) indeed emerges from (ωμ, ω0 i), endowed ab initio with intertwined tangent and cotangent Clifford algebras. A deeper level of Kähler calculus, i.e., the language of Dirac equations, thus emerges. This makes the existance of an intimate relationship between classical differential geometry and quantum theory become ever more plausible. The issue of a geometric canonical Dirac equation is also raised. (shrink)

In a recent paper [J. G. Vargas and D. G. Torr, Found. Phys. 27, 599 (1997)], we have shown that a subset of the differential invariants that define teleparallel connections in spacetime generates a teleparallel Kaluza-Klein space (KKS) endowed with a very rich Clifford structure. A canonical Dirac equation hidden in this structure might be uncovered with the help of a teleparallel Kähler calculus in KKS. To bridge the gap to such a calculus from the existing Riemannian Kähler calculus in (...) spacetime, we commence the construction of a teleparallel Kähler calculus in spacetime. In the process, we notice: (a) Unknown to him, one of Einstein's equations in his attempt at unification with teleparallelism states that the interior covariant derivative of the torsion is zero. (b) A mechanism exists in the tangent bundle of teleparallel spaces for producing confinement (in the applicable cases, one would have to show why nonconfinement also occurs, rather than the other way around). (c) When the torsion is not zero, the interior covariant derivative in the sense of Kähler, δF, does not coincide with *d*F. The system (dF = 0, δF = j) rather than (dF = 0, *d*F = j) should then be used for generalizations of Maxwell's electrodynamics. (shrink)

Adequality, or παρισóτης (parisotēs) in the original Greek of Diophantus 1 , is a crucial step in Fermat’s method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s collected works (1891, pp. 133–172). The first article, Methodus ad Disquirendam Maximam et Minimam 2 , opens with a summary of an algorithm for finding the maximum or minimum value of an (...) algebraic expression in a variable A. For convenience, we will write such an expression in modern functional notation as f (a). 3 The algorithm can be broken up into six steps in the following way:Introduce an auxiliary .. (shrink)

Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporary (...) geometric morphometrics, the notion of a morphospace is based on the Euclidean tangent space to Kendall’s shape space, which is a Riemannian manifold. Many more classical morphospaces, such as Raup’s space of coiled shells, lack these metric properties, e.g., due to incommensurably scaled variables, so that these morphospaces typically are affine vector spaces. Other notions of a morphospace, like Thomas and Reif’s skeleton space, may not give rise to a quantitative measure of similarity at all. Such spaces can often be characterized in terms of topological or pretopological spaces. (shrink)

Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers (...) did : “geometrization of arithmetic”..Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him, the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes. (shrink)

We consider quantum mechanics written in hydrodynamic formulation for the case of relativistic spinor fields to study their velocity: within such a hydrodynamic formulation it is possible to see that the velocity as is usually defined can not actually represent the tangent vector to the trajectories of particles. We propose an alternative definition for this tangent vector and hence for the trajectories of particles, which we believe to be new and the only one possible. We discuss how these (...) results are a necessary step to take in order to face further problems, like the definition of trajectories for multi-particle systems or ensembles, as they happen to be useful in many applications and interpretations of quantum mechanics. (shrink)

L'échapper belle signifie "se sortir de justesse d'une situation délicate". Cependant, prise au pied de la lettre, l'expression pourrait signifier "s'extraire d'une impasse d'une belle façon". Il s'agirait alors de la capacité à trouver des façons efficaces et élégantes d'échapper à des situations épineuses. Si l'échappée peut évoquer la fuite, voire la lâcheté, l'échappée belle relève du courage et du panache. Quelle sagesse recèle cette capacité à prendre la tangente en mariant le cran et le style? Comment rendre l'échappée belle? (...) Cet essai examine la notion d'échappée belle sous ses différentes facettes en faisant appel à des exemples empruntés à la littérature, à l'histoire et à la pop culture. Inventivité et créativité, esquive et ruse, manipulations lexicales, voies détournées, subtilités et subterfuges, tout est permis pour qui veut l'échapper belle. (shrink)

In this paper, we give a thorough account of Brianchon and Poncelet’s joint memoir on equilateral hyperbolas subject to four given conditions, focusing on the most significant theorems expounded therein, and the determination of the “nine-point circle”. We also discuss about the origin of this very rare example of collaborative work for the time, and the general challenge of finding the nature of the loci described by the centres of the conic sections required to pass through m points and to (...) be tangent to n straight lines given in position, m + n = 4, which was posed at the end of their work. In the case m = 4, i.e. when the conic sections have to pass through the vertices of a quadrilateral, the locus of centres is another conic section passing through the intersection points of the opposite sides and the two diagonals of the quadrilateral, respectively, and, as Gergonne showed analytically shortly after, through other significant points connected with the quadrilateral; this curve was later given the name of the “nine-point conic”, being a natural generalization of the above mentioned circle. (shrink)

This chapter discusses the problem of time in the thought of Bergson, showing how the evolution of the concept of duration is conducive to new developments in the philosophy of intuition. Duration, which in the Essay connotes the experience of a non-measurable lived experience, while in Matter and Memory it assumes rhythms of different intensities to justify the relationship between perception and memory, as well as in Creative Evolution is judged as the fabric of reality itself, in Duration and Simultaneity (...) it is posited not only as a criterion to discern what is real and what is artificial, but also to justify the measurement of reality, that is to say, to restore the point of contact between time as duration and space, that seemed previously compromised by the loss of ontological consistency of extension as the dimension of the body. In this paper, then, Bergson, also by virtue of a controversial, detailed comparison with the theory of relativity, finally arrives to support the hypothesis of a single temporal dimension, curved and dynamic, where space comes to be outlined as the abstract tangent of time. It is at this level, then, that it becomes possible to examine the emergence of the concepts within the meanders of intuition as concepts imbued with meaning: here is the primary source of that continuous “added” element of new nuclei of creativity that characterizes the very logic of Bergsonian living where what, as ideal relationship, is time, as a real relationship, becomes life, i.e., the continuous opening of a register in which time is inscribed, becoming incarnate thus in the very nature of every organism. (shrink)