Didactical situations to treat Lorentz and Galileo transformations in theoretical physics

Notions of Electromagnetism and Special Theory of Relativity (STR) require important mathematical knowledge applied to theoretical physics. Recognizing pedagogical difficulties in the teaching of theoretical physics, the Theory of Didactical Situations (TDS), which consists of a set of practices that aim to contribute to the improvement of mathematics teaching. In this context, the present work is motivated to present a set of practices based on TDS with a focus on teaching Electromagnetism and STR, where problems that require an understanding of the transformations of Galileo and Lorentz. Specifically, the didactic situation is constructed by means of four problem proposals, while in the adidatic situation, the student is invited to understand the roles of these transformations in the study of these problems. Ultimately, the relevance of the educator in the institutionalization situation is reinforced, a moment when it must be clarified how all mathematical relations are strongly related to physical principles.

Extraction of New Applicable Dimensionless Relations in the Wedge Impact Problem Using WCSPH Method

Impact problem associated with water entry of a wedge has important applications in various aspects of naval architecture and ocean engineering. In the present study, the 2DOF (2 Degrees of Freedom) wedge impact problem into the water with various wedge deadrise angles and impact velocities is investigated using Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. Artificial viscosity and density correction are used to create stability and also to prevent the penetration of fluid particles into the solid boundary. Solving the impact problem is very time-consuming, therefore extracting new mathematical relations can be very useful to calculate some important and applicable parameters in a certain range of wedge angles and impact velocities. In the present research, some new dimensionless applicable relations using the Buckingham π theorem are extracted to investigate important parameters such as acceleration and slamming force in general cases of a wedge impact problem. Then, these mathematical relations are validated by the results obtained from the simulations.

Understanding the index theorems with massive fermions

The index theorems relate the gauge field and metric on a manifold to the solution of the Dirac equation on it. In the standard approach, the Dirac operator must be massless to make the chirality operator well defined. In physics, however, the index theorem appears as a consequence of chiral anomaly, which is an explicit breaking of the symmetry. It is then natural to ask if we can understand the index theorems in a massive fermion system which does not have chiral symmetry. In this review, we discuss how to reformulate the chiral anomaly and index theorems with massive Dirac operators, where we find nontrivial mathematical relations between massless and massive fermions. A special focus is placed on the Atiyah–Patodi–Singer index, whose original formulation requires a physicist-unfriendly boundary condition, while the corresponding massive domain-wall fermion reformulation does not. The massive formulation provides a natural understanding of the anomaly inflow between the bulk and edge in particle and condensed matter physics.

ROLE AND IMPORTANCE OF STHIRA BINDU (FIXED POINT) IN YOGA PHILOSOPHY

It is true that all living things and all the mechanisms of entire universe are guided by mathematical relations and results. The theory of fixed point is one of the most leading gears of modern mathematics and its results are the most generally useful in mathematics which gives the solution of non-linear problems of various fields of modern subjects [9]. Also, the human brain can perform the intellectual courses that still have not been performed by digital computers. It may therefore be seen that quantum mechanics is very much associated with the consciousness of mankind [21]. Yoga is one of the few ways to understand the eventual reality mentioned in Vedanta, quantum physics and mathematics as well [19]. This paper investigates the role and importance of fixed point in eastern philosophies especially with yoga along with meditation focusing that mathematics plays a significant role in yoga philosophy.

Analysis of Distortion Based on 2D MEMS Micromirror Scanning Projection System

To analyze the distortion problem of two-dimensional micro-electromechanical system (MEMS) micromirror in-plane scanning, this paper makes a full theoretical analysis of the distortion causes from many aspects. Firstly, the mathematical relations among the deflection angle, laser incidence angle, and plane scanning distance of the micromirror are constructed, and the types of projection distortion of the micromirror scanning are discussed. Then the simulation results of reflection angle distribution and point cloud distribution are verified by MATLAB software under different working conditions. Finally, a two-dimensional MEMS micromirror scanning projection system is built. The predetermined waveform can be scanned and projected successfully. The distortion theory is proved to be correct by analyzing the distortion of the projection images, which lays a foundation for practical engineering application.

W poszukiwaniu rozumowania matematycznego najmłodszych uczniów – doniesienie z badań wdrożeniowych

Mathematical reasoning is a crucial competence for the construction of useful knowledge. The text presents selected results of the study of students of the third grade of an elementary school in the field of text task solving. The conducted educational experiment showed the potential of solving non-standard tasks for developing the reasoning of mathematically weaker students. Contact with tasks that require independent mathematizing allowed to reduce the number of students with the lowest results in the post test. The ways of understanding the role of drawing in exploring mathematical relations described in the task were analysed.

Mathematical relations for constructing an algorithm for estimating signal parameters under conditions of constraints

Статья посвящена разработке математических соотношений для построения алгоритма оценивания параметров сигналов в условиях ограничений. При работе транспортной системы возникают довольно сложные проблемы, которые связаны с необходимостью проведения оценки принятых параметров с требованиями соблюдения имеющихся ограничений. Ограничения могут представлять собой как равенства, так и неравенства. Поскольку ограничения-неравенства могут быть сведены путём добавления фиктивных переменных к условиям, а также их можно проверить по шагам, переводя в состав равенства, в статье разработан алгоритм, позволяющий иметь ограничения-равенства. Данная задача относится к классу статистических проблем оптимизации. Для ее решения использованы стандартные функции из подкаталога "optimization" вычислительной среды MatLAB. Построение такого алгоритма даст возможность не только уменьшить складские расходы, но и сократить основное производственное время. The article is devoted to the development of mathematical relationships for constructing an algorithm for estimating signal parameters under constraints. During the operation of the transport system, rather complex problems arise, which are associated with the need to assess the adopted parameters with the requirements of compliance with the existing restrictions. Constraints can be either equality or inequality. Since the inequality constraint can be reduced by adding dummy variables to the equality conditions, and they can also be checked step by step, transforming them into equality, we will develop an algorithm that allows us to have equality constraints. This task belongs to the class of statistical optimization problems. To solve it, standard functions from the "optimization" subdirectory of the MatLAB computing environment will be used. The construction of such an algorithm will make it possible not only to reduce storage costs, but also to reduce the main production time.

DYNAMICS OF SEEDLING PLANTING MACHINE EQUIPPED WITH VERTICAL DISTRIBUTOR AND ARTICULATED BUCKETS PLANTING UNIT

In this work we make a comparative analysis of the dynamics for two variants of seedling planting machines equipped with distributors and articulated buckets respectively: with the seedlings being placed in the furrow opened by a coulter; with the seedling being planted directly into the ground. The dynamics of the seedling planting machine equipped with vertical distributors and articulated buckets is seen from the perspective of the working process involving the placement of the seedling in the ground, its release, its covering with earth and its additional compaction. In principle, furrow opening is performed by a coulter and the vertical distributor with articulated buckets places and releases the seedling in the furrow, in the first variant, and in the second one, the seedling is inserted and released directly into the ground. In the paper are written the mathematical relations describing the dynamics of the seedling planting machine, in the two variants, and we perform their testing, make recordings, interpret the results, reach conclusions and make recommendations on the optimum variant. The agro-technical parameters and tensile strength of the planting machine are analysed in the two functional variants. The experiments were performed under the same working conditions for both variants analysed.

Assessment of stand-alone photovoltaic system and mini-grid solar system as solutions to electrification of remote villages in Afghanistan

Afghanistan enjoys huge renewable energy, especially solar resources. Meanwhile, most of the population especially people who live in remote rural areas, still do not have appropriate access to electricity. Poor access to energy has made life more challenging and deprived rustic people from related primary living facilities. To address this grant challenge, considering the high potential of solar energy available in the country, this paper presents a study on design and economic comparison of the two most feasible methods of solar power production for rural areas in Afghanistan. In the first method, a stand-alone Solar Photovoltaic (PV) system has individually been considered in every single house of a village. In this way, energy is produced and consumed in each house itself. While in the second method, energy for the whole village is produced by a micro solar power station in a centralized manner and then distributed through a 0.4 kV islanded Mini grid all around the village. The study is carried out through conventional mathematical relations, based on daily energy demand in a rural household. The result indicates that implementation of the second method is not only best affordable but also more viable and will create other socio-economic opportunities.

The Poynting Vector and the New Theory of a Transformer. Part 11. Three-Phase Three-Core Transformers without a Neutral Wire

A topological equivalent circuit for a three-phase three-core transformer reflecting the spatial structure of its magnetic system is developed. Owing to this approach, it became possible to represent the magnetic fluxes of the magnetic circuit’s all main sections and the apertures for each of three phases directly in the circuit in the absence of the windings’ neutral wires. The circuit is constructed by stitching together the anatomical circuit models of single-phase transformers obtained in the previous parts with taking into account the relationships between the fluxes at the junctions of the phase zones in iron. Its validity is confirmed by the rigor nature of the physical and mathematical relations for idealized transformers with infinite magnetic permeability of iron and simplified magnetic field patterns, which corresponds to the generally accepted approach with neglecting the magnetization currents. The difference lies in the fact that the developed model takes into account the heterogeneity of magnetization in different parts of the magnetic circuit with allocating more than 30 sections in the iron and apertures. The transition to the model of a real three-core transformer is carried out by adding four nonlinear transverse magnetization branches in each extreme phase zone and eight branches in the central phase zone to the idealized equivalent circuit. It is shown that in cases of winding connections without neutral wires, there is no flux of the Poynting vector in interphase zones in any unbalanced mode. In this case, the problems connected with the occurrence of fluxes exceeding the no-load fluxes under the conditions of symmetric and asymmetric short circuits, as well as the occurrence of buckling fluxes in these modes in the region outside the transformer iron, are solved.