D’Alembert’s Contribution to the French Enlightenment

The French Enlightenment directly influenced and promoted the Enlightenment in other European countries. During the Enlightenment, the development of natural science and the dissemination of scientific knowledge greatly promoted the emancipation of human minds. D’Alembert is a famous French mathematician, physicist, astronomer, and philosopher. As a representative on mission during the French Enlightenment, d’Alembert made important contributions to mechanics, mathematics, and astronomy that greatly promoted the development of natural sciences.

3. Beautiful theories, ugly facts

‘Beautiful theories, ugly facts’ evaluates the theories on planetary systems, particularly the Solar System. In 1734, the Swedish polymath Emmanuel Swedenborg proposed that the Sun and all the planets condensed out of the same ball of gas, in what is probably the earliest statement of the nebular hypothesis. The nebular hypothesis entered something close to its modern form in the hands of the French mathematician Pierre-Simon Laplace, who in 1796 made the clear connection to Newtonian gravity. The angular momentum problem and the structure of a protoplanetary disk, the formation of rocky cores, and the gravitational accretion of gas in the disk also come under this topic.

Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form QM(u) = x^2 + y^2 + z^2 - c^2 t^2where (x, y, z) are the spacial components of the vector u and t is the time component of u and c is the constant of the speed of light. In this Minkowski-Lorentz space, a Dupin cyclide is the union of two conics on the unit pseudo-hypersphere, called the space of spheres, and a singular point of a Dupin cyclide is represented by an isotropic vector. Then, we model Dupin cyclides using rational quadratic Bézier curves with mass points. The subdivisions of a surface i.e. a Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space ε3, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are circles which look like ellipses.

Cartesius dissoluto / Dissolute Cartesius

Resumo: No romance Catatau, de Paulo Leminski, uma personagem reconhecível se projeta: Renatus Cartesius, produtor do imenso solilóquio que constitui a totalidade do texto, parece corresponder a René Descartes, o famoso matemático francês do século XVII. Postado sob uma árvore do Jardim Botânico de Recife, entre as lentes de sua luneta e o cachimbo de erva narcótica que vorazmente aspira, Cartesius toma contato com a selvagem natureza brasileira, ainda que (cartesianamente) organizada sob as formas de um jardim zoobotânico. Este artigo pretende investigar, a princípio, o modo pelo qual o romance- ideia de Leminski agencia esse inevitável contraste entre dois Descartes: o Descartes da história da filosofia, autor das Regras para a direção do espírito, e o desregrado Cartesius da situação ficcional engendrada por Leminski. Para realizar tal investigação, este artigo procurou se ater a algumas passagens do romance, com o intuito de checar os modos pelos quais a paródica e carnavalesca figuração de Descartes se baseia em uma anticartesiana dissolução da integridade egoica da personagem, gerando, com isso, muito mais que superficial humorismo. Antes, o artigo defende a hipótese de que, com a dissolução da figura de Descartes, o romance esboça antropofágica reação aos processos coloniais a que o Brasil teria sido exposto, ao longo de sua história. Tal reação, contracolonizadora, se sustenta na medida em que associa o esfacelamento egoico de Cartesius ao impacto causado por uma exuberante natureza que, por sua hiperbólica constituição, não se submete às quadraturas impostas pelo pensamento europeu – de que Descartes se faz emblema.Palavras-chave: Paulo Leminski; Catatau; René Descartes; literatura brasileira; estudo de personagem; filosofia.Abstract: In Paulo Leminski’s novel, Catatau, a recognizable character is projected: Renatus Cartesius, the immense soliloquy’s producer that constitutes the entire text, seems to correspond to René Descartes, the famous 17th century French mathematician. Standing under a tree at the Recife Botanical Garden between the lenses of his bezel and the narcotic weed pipe he voraciously smokes, Cartesius makes contact with the wild Brazilian nature, although (Cartesian) organized in the form of a zoobotanical garden. This article intends to investigate at first the way in which Leminski’s novel-idea handles this inevitable contrast between both Descartes: Descartes from the History of Philosophy, author of Rules for the Direction of the Mind and the intemperate Cartesius from the fictional situation engendered by Leminski. To conduct such an investigation, this article sought to stick to some passages of the novel in order to check the ways in which Descartes’s parody and carnival figuration are based on an anticartesian dissolution of the character’s egoic integrity, thus generating much more than superficial humor. Rather, the article defends the hypothesis that with the dissolution of Descartes’s figure, the novel outlines an anthropophagic reaction to the colonial processes to which Brazil would have been exposed throughout its history. This counter-colonizing reaction is sustained insofar as it associates Cartesius’s egoic disintegration with the impact caused by an exuberant nature, which due to its hyperbolic constitution, does not submit to the quadratues imposed by the European thought – of which Descartes becomes an emblem.Keywords: Paulo Leminski; Catatau; René Descartes; Brazilian literature; character study; philosophy.

FROM THE HISTORY OF THE BINARY NUMBER SYSTEM. UNKNOWN AUTHOR'S BROCHURE

The article describes the first in history, according to the author, "practical use of the binary number system". This is an old puzzle, in different languages having different names, first described in the brochure "Théorie du baguenodier..." ("Theory of baguenodier...") published in France in 1872. The diagrams in the brochure show the various puzzle states and binary state numbers. French mathematician Éduard Lucas developed the idea of using the connection of a situation on a diagram that simulates a puzzle with the binary number of the corresponding move to determine the number of moves required to move from any one state of the puzzle to any other. This number is equal to the difference between the decimal equivalents of the binary numbers of the two states. In addition, Lucas pointed out the relationship between the binary numbers of all states of a puzzle with n rings with all possible combinations without repetitions of n objects.

Henri Poincare’s legacy for tipping points

<p>The science of Earth system and climate tipping points has evolved and matured as a disciplined approach to understanding anthropogenic and non-anthropogenic stresses on the Earth’s subsystems in the 21<sup>st</sup> century. However, tipping points is strongly interlinked with the science of bifurcations and dynamical systems, which received a seminal and resonant illumination by the great French mathematician Henri Poincare (1854-1912). Thus, quite a few historically minded tipping point scientists mention Poincare as the seminal, path-setting thinker for tipping point understandings.</p><p>Moreover, Poincare’s bifurcation and dynamical systems-pertinent science is also linked to his seminal role in chaos theory, which illuminates today’s understanding of climate stochasticity. Poincare famously said, "A very small cause which escapes us determines a considerable effect that we cannot see; so, we say this effect is random," which provided grounding for the chaos notion of critical sensitivity to initial conditions. Since Poincare, great strides in abrupt change understanding as linked to chaos (and within an examination of turbulence) have taken place in the science that informs tipping points, such as with the work of Ed Lorenz and David Ruelle. Additionally, the Russian mathematicians (e.g., Andronov and Arnold) have contributed greatly with the refining of differential equations for bifurcation understandings that Poincare began.  </p><p>This EGU presentation is a history of science presentation on Henri Poincare's commencement of bifurcation, dynamical system and chaos understandings as presented by a journalist who has done both interviews and general historical research. The presentation sets key points in Poincare’s biography and pertinent career and sketches the legacy of this Poincare focus up from Henri Poincare through Russian bifurcation scientists, catastrophe theorist Rene Thom, and ultimately Lorenz and current bifurcation theorists, such as Michael Ghil and Valerio Lucarini. It offers light on the ancestry of one of the most important examinations of the Anthropocene, climate change tipping points.  </p>

A Note on Anti-Berge Equilibrium for Bimatrix Game

Game theory plays an important role in applied mathematics, economics and decision theory. There are many works devoted to game theory. Most of them deals with a Nash equilibrium. A global search algorithm for finding a Nash equilibrium was proposed in [13]. Also, the extraproximal and extragradient algorithms for the Nash equilibrium have been discussed in [3]. Berge equilibrium is a model of cooperation in social dilemmas, including the Prisoner’s Dilemma games [15]. The Berge equilibrium concept was introduced by the French mathematician Claude Berge [5] for coalition games. The first research works of Berge equilibrium were conducted by Vaisman and Zhukovskiy [18; 19]. A method for constructing a Berge equilibrium which is Pareto-maximal with respect to all other Berge equilibriums has been examined in Zhukovskiy [10]. Also, the equilibrium was studied in [16] from a view point of differential games. Abalo and Kostreva [1; 2] proved the existence theorems for pure-strategy Berge equilibrium in strategic-form games of differential games. Nessah [11] and Larbani, Tazdait [12] provided with a new existence theorem. Applications of Berge equilibrium in social science have been discussed in [6; 17]. Also, the work [7] deals with an application of Berge equilibrium in economics. Connection of Nash and Berge equilibriums has been shown in [17]. Most recently, the Berge equilibrium was examined in Enkhbat and Batbileg [14] for Bimatrix game with its nonconvex optimization reduction. In this paper, inspired by Nash and Berge equilibriums, we introduce a new notion of equilibrium so-called Anti-Berge equilibrium. The main goal of this paper is to examine Anti-Berge equilibrium for bimatrix game. The work is organized as follows. Section 2 is devoted to the existence of Anti-Berge equilibrium in a bimatrix game for mixed strategies. In Section 3, an optimization formulation of Anti-Berge equilibrium has been formulated.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

When Is a Proof a Proof?

In his famous talk at ICME 2 (Exeter 1972) the French mathematician R. Thom pointed out that any conception of mathematics teaching necessarily rests on a certain view of mathematics (Thom 1973, 204). As a consequence mathematics education cannot develop without close links to mathematics.