Dual Approach in the Application of Geometric Interpretation of Linear Programming on the Organisation of Goods Distribution

The topic of the paper is the application of dual approach in formulation and resolution of goods distribution tasks problems. The gap in previous goods distribution research is the absence of the methodologies and goods transportation calculation methods for manufacturing companies with not too large amount of goods distribution whereby goods distribution is not the core activity. The goal of this paper is to find a solution for transportation in such companies. In such cases it is not rational to procure a specific software for the improvement of goods transportation but rather apply the calculation presented in this paper. The aim of this paper from mathematical aspect is to show the convenience of switching from the basic geometric interpretation of linear programming applied on transportation tasks to dual approach for the companies with too many costs limitations per transport task but not enough available transportation means. Recent research studies that use dual approach in linear programming are generally not applied to transportation tasks although such approach is very convenient. The goal of the paper is also to resolve transportation tasks by both primal and dual approach in order to prove the correctness of the method.

NAJSTARSZE (XI-XII W.) FIGURY SZACHOWE WYKONANE W STYLISTYCE ABSTRAKCYJNEJ ARABSKIEJ Z TERENU POLSKI , NA TLE TRENDÓW EUROPEJSKICH ZWIĄZANYCH Z POCZĄTKOWYMI FAZAMI ADAPTACJI W EUROPIE

Chess is a board game, in the Middle Ages referred to as a tabula. During the long way it took since its origin in India in the 6th century until modern times, the subsequent communities left their own, inimitable cultural marks. In India, chess had a deeply mystical nature; Persians used chess to picture the world as a battlefield; Arabs systematised many concepts and took note of the mathematical aspect; Europe made use of chess to define rules that should apply to an ideal society. This shows a perfect understanding of the balance on the chessboard, the mutual dependencies and consistent actions leading to success – both when playing and creating social life. Medieval literature provides an excellent basis for studies of the intertwining cultural trends and describing the reality. In the literature, elements based on playing chess are oftentimes among the postulated modes of education. However, the ideas encountered by the potential users of chess tournaments were best communicated by the figures and the accumulated plethora of notions. An analysis of the changes affecting jackstraws at an early stage of the game’s adaptation in Europe and other territories which took over chess as cultural models, leads to a conclusion that the material from the 11th-12th centuries that comes from Polish collections matches many Latin trends and shows considerable knowledge thereof.

Albert Einstein’s Epistemic Virtues and Vices

The article is based on the concepts of epistemic virtues and epistemic vices and explores A. Einstein’s contribution to the creation of fundamental physical theories, namely the special theory of relativity and general theory of relativity, as well as to the development of a unified field theory on the basis of the geometric field program, which never led to success. Among the main epistemic virtues that led Einstein to success in the construction of the special theory of relativity are the following: a unique physical intuition based on the method of thought experiment and the need for an experimental justification of space-time concepts; striving for simplicity and elegance of theory; scientific courage, rebelliousness, signifying the readiness to engage in confrontation with scientific conventional dogmas and authorities. In the creation of general theory of relativity, another intellectual virtue was added to these virtues: the belief in the heuristic power of the mathematical aspect of physics. At the same time, he had to overcome his initial underestimation of the H. Minkowski’s four-dimensional concept of space and time, which has manifested in a distinctive flexibility of thinking typical for Einstein in his early years. The creative role of Einstein’s mistakes on the way to general relativity was emphasized. These mistakes were mostly related to the difficulties of harmonizing the mathematical and physical aspects of theory, less so to epistemic vices. The ambivalence of the concept of epistemic virtues, which can be transformed into epistemic vices, is noted. This transformation happened in the second half of Einstein’s life, when he for more than thirty years unsuccessfully tried to build a unified geometric field theory and to find an alternative to quantum mechanics with their probabilistic and Copenhagen interpretation In this case, we can talk about the following epistemic vices: the revaluation of mathematical aspect and underestimation of experimentally – empirical aspect of the theory; adopting the concepts general relativity is based on (continualism, classical causality, geometric nature of fundamental interactions) as fundamental; unprecedented persistence in defending the GFP (geometrical field program), despite its failures, and a certain loss of the flexibility of thinking. A cosmological history that is associated both with the application of GTR (general theory of relativity) to the structure of the Universe, and with the missed possibility of discovering the theory of the expanding Universe is intermediate in relation to Einstein’s epistemic virtues and vices. This opportunity was realized by A.A. Friedmann, who defeated Einstein in the dispute about if the Universe was stationary or nonstationary. In this dispute some of Einstein’s vices were revealed, which Friedman did not have. The connection between epistemic virtues and the methodological principles of physics and also with the “fallibilist” concept of scientific knowledge development has been noted.

Textbooks using mathematics in the margins of current mathematics: manuals for teaching inheritance in Morocco

Nowadays in Morocco, the distribution of inheritance is part of the basic juridical-religious curriculum taught in high school and in some branches of humanities and social sciences in higher education. This question of inheritance is, in fact, a taught discipline - called cilm al-farā'iḍ (lit. science of prescriptions) - with its own textbooks and its own teachers. In their presentation of the mathematical aspect related to this question of inheritance, these textbooks implement mathematics which uses a language and a vocabulary specific and different from those conveyed in modern mathematical teaching. In this paper we try to shed some light on this mathematical practice and highlight some of its peculiarities. For this, we present some textbooks for the teaching of this discipline by focusing on their mathematical content. We will also highlight the basic features of mathematics conveyed by these textbooks and their differences with current mathematics. Keywords: Inheritance problems, vocabulary, fraction, resolution process, manuals.