Apontamentos iniciais acerca de uma carta escrita por al-Biruni (973-1048) no século XI

O trabalho ora apresentado se concentra na área de História da Matemática, mais precisamente na História da Matemática do Islã medieval. Discutiremos informações levantadas sobre uma carta escrita por al-Biruni que diz respeito a Lei dos Senos na Trigonometria Triangular Esférica, trataremos de seu conteúdo e estrutura, bem como observaremos a forma de registro do conhecimento na tentativa de responder à pergunta que norteia a construção do presente artigo: o que pode se falar sobre a estrutura da carta escrita por al-Biruni e que aspectos observados na carta, podem nortear a construção de uma narrativa que se aproxime de uma perspectiva atualizada de registrar a História? Usaremos para o desenvolvimento do trabalho historiográfico uma tradução da carta que está em inglês e foi encontrada no livro Beyruni’ye Armagan, mais precisamente no capítulo intitulado Al Beyruni's Letter on Abu Nasr Mansur's Demonstration of the Sine Law que ocupa as páginas 183 a 197 e tem autoria do Dr. Aydin Sayli. No que diz respeito a carta, a original está em árabe e o livro supracitado consta, ainda, uma tradução turca. A partir do estudo da versão inglesa, em confronto com demais versões presentes no livro, verificamos que ela trata de seis casos da lei dos senos, diluídos em quatro demonstrações, os três primeiros casos são associados a Trigonometria Triangular Esférica (TTE) enquanto os últimos três, são associados a Geometria Plana. Assim, nosso objetivo é apresentar parte de uma análise historiográfica (ainda em desenvolvimento) e, pretendemos alcançá-lo pelo desenvolvimento desse trabalho que, mediante a questões metodológicas, se associa a uma pesquisa com abordagem qualitativa e de procedimentos documentais e bibliográficos. Como resultado, apresentamos uma apreciação preliminar da carta e possíveis apontamentos que podem nortear a construção de uma narrativa mais contextualizada sobre o documento. Palavras-chave: al-Biruni; carta sobre a lei dos senos; Trigonometria Triangular Esférica.

An Integrative Model of Plant Gravitropism Linking Statoliths Position and Auxin Transport

Gravity is a major cue for the proper growth and development of plants. The response of plants to gravity implies starch-filled plastids, the statoliths, which sediments at the bottom of the gravisensing cells, the statocytes. Statoliths are assumed to modify the transport of the growth hormone, auxin, by acting on specific auxin transporters, PIN proteins. However, the complete gravitropic signaling pathway from the intracellular signal associated to statoliths to the plant bending is still not well-understood. In this article, we build on recent experimental results showing that statoliths do not act as gravitational force sensor, but as position sensor, to develop a bottom-up theory of plant gravitropism. The main hypothesis of the model is that the presence of statoliths modifies PIN trafficking close to the cell membrane. This basic assumption, coupled with auxin transport and growth in an idealized tissue made of a one-dimensional array of cells, recovers several major features of the gravitropic response of plants. First, the model provides a new interpretation for the response of a plant to a steady stimulus, the so-called sine-law of plant gravitropism. Second, it predicts the existence of a gravity-independent memory process as observed recently in experiments studying the response to transient stimulus. The model suggests that the timescale of this process is associated to PIN turnover, calling for new experimental studies.

Estimating sea vessel’s braking distance influenced by wind and waves

The article considers the methods of estimating the wind effect on the braking characteristics and the vessel behavior under the wind and waves. A model with regular sea waves is illustrated, in which the wave ordinate changes according to the sine law. It is stated that in reality the wave ordinate varies approximately by the sum of an infinite number of harmonics, the so-called irregular excitation. There are given differential equations of wind effect on the inertial-braking characteristics, of braking, of calculating the braking distance, etc. There has been estimated the braking distance of the motor vessel “Yuri Maksarev” with four rows of containers in headwinds up to 15 m/s. The wind has been proved to influence greatly the length of the braking distances, with the length of braking distance becoming two cables less after three minutes. The diagram of the apparent wind influencing the ship is presented. The apparent wind impacts the vessel’s hull above the waterline. Drift angle estimating is performed using a programmable calculator, a table or a nomogram. There is proposed a nomogram that greatly simplifies preliminary calculating and reduces the time costs. There have been compiled the programs for calculating the wind drift angle and constructing the corresponding nomogram in the application Microsoft Excel 97 using the programming language Visual Basic. The developed programs allow automating the most frequently performed calculations for determining the angle of wind drift, for which little time is needed during the maneuvering process. The arrangement of the diagram on the paper is illustrated, the process of building a nomogram is presented in stages. A case of the calculated nomogram for the motor ship “Yuri Maksarev” has been presented.

An integrative model of plant gravitropism linking statoliths position and auxin transport

ABSTRACTGravity is a major cue for proper plant growth. The response to this cue implies starch-filled plastids, the statoliths, sedimenting on the bottom of statocytes cells. These statoliths are assumed to modify the transport of the growth hormone by acting on specific hormone carriers, PIN proteins, in an unknown way. Recent experiments show that statholiths do not act as gravitational force sensor, but as position sensor. Moreover, the signal is not immediate but integrated over a time-scale of a tens of minutes. However, the precise gravitropic signaling pathway from the sensing of this cue at the cell scale to the response in growth at the tissue scale is still not understood. Here we present a bottom-up theory that enables to rationalize the previous phenomenological results from microscopic considerations. Our approach, consistent with existing experimental results, tends to support that the integration time-scale is associated to PIN turnover. Moreover, it leads to a revision of the so-called sine-law of plant gravitropism.

An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space

We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process X t with one parameter t. If there is another varying key factor of an event described by a process, then we need to consider another parameter besides t. That is, we need a system of random variables with two parameters, say X s , t , which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes.

Quenching of Aluminum Solid Cylinder: Numerical Study

A numerical method is presented for evaluating the residual stress distribution in a long aluminum solid cylinder subjected to rapid cooling. An analytical model is developed for the temperature distribution. For the boundary conditions, experimental data for the outer surface of the cylinder are used, and a reasonable agreement between the predicted temperature distribution at the center of the cylinder and the experimental data is observed. For the numerical analysis, a quasi-static, uncoupled thermoelastoplastic analysis, based on a hyperbolic sine law, is presented. The numerical results are presented for the temperature distribution as well as the thermoelastoplastic stress distribution in a solid cylinder with temperature-dependent properties. The residual stress distribution is compared with the results of other investigators who used the Finite Element Method, and a reasonable agreement between our results and previous results is observed. The conclusion is reached that the temperature dependency of the yield stresses and the problem of post-yielding are two important factors to be considered when developing a model for predicting the residual stresses in quenched bodies.