Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system

Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form. However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors. Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown. By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loading

Two-Dimensional Cartesian Coordinate System Educational Toolkit: 2D-CACSET

Engineering education benefits from the application of modern technology, allowing students to learn essential Science, Technology, Engineering, and Mathematics (STEM) related concepts through hands-on experiences. Robotic kits have been used as an innovative tool in some educational fields, being readily accepted and adopted. However, most of the time, such kits’ knowledge level requires understanding basic concepts that are not always appropriate for the student. A critical concept in engineering is the Cartesian Coordinate System (CCS), an essential tool for every engineering, from graphing functions to data analysis in robotics and control applications and beyond. This paper presents the design and implementation of a novel Two-Dimensional Cartesian Coordinate System Educational Toolkit (2D-CACSET) to teach the two-dimensional representations as the first step to construct spatial thinking. This innovative educational toolkit is based on real-time location systems using Ultra-Wide Band technology. It comprises a workbench, four Anchors pinpointing X+, X−, Y+, Y− axes, seven Tags representing points in the plane, one listener connected to a PC collecting the position of the Tags, and a Graphical User Interface displaying these positions. The Educational Mechatronics Conceptual Framework (EMCF) enables constructing knowledge in concrete, graphic, and abstract levels. Hence, the students acquire this knowledge to apply it further down their career path. For this paper, three instructional designs were designed using the 2D-CACSET and the EMCF to learn about coordinate axes, quadrants, and a point in the CCS.

Abstraction in Archaeological Stratigraphy: a Pyrenean Lineage of Innovation (late 19th-early 21th century)

Methodological innovations have a special status in disciplinary histories, because they can be widely adopted and anonymised. In the 1950s, this occurred to Georges Laplace’s innovative use of 3-dimensional metric Cartesian coordinate system to record the positions of archaeological objects. This paper proposes a conceptual and social history of this process, with a focus on its spatial context, the Pyrenean region (Spain, Basque Country, and France). Main results of this research based on archives, publications, and bibliometric data, include: 1) a critical discussion of the notions concerning authorship of such methodological innovations; 2) a presentation of the lesser-known aspects of Laplace’s method, showing its contribution to the abstraction and formalisation of archaeological observations and data recording; and 3) the identification of an international Pyrenean intellectual lineage of innovation regarding stratigraphy and excavation methods, from the late 19th century to the early 21th century.

Adaptive technology for constructing the kinetic equations of reduction reactions under conditions of a priori uncertainty

The object of research is the process of oxide reduction in a reaction system of mass m due to the reaction on a contact surface with an area of S. An adaptive technology is proposed that allows one to construct the kinetic equation of the process in which the oxide is reduced from the initial product under conditions of a priori uncertainty. A priori uncertainty regarding the behavior of a physicochemical system is understood as the fact that the following information is not available to the researcher: – about the change in the mass of the reaction system and the area of the contact surface; – about the rate of accumulation of the finished product; – about the time of withdrawal of the finished product from the system. The proposed adaptive technology includes five sequential stages to eliminate a priori uncertainty. This is ensured through the use of an adaptive algorithm, which allows obtaining the maximum accuracy in estimating the output variable by selecting the optimal parameter of the adaptive algorithm, and the subsequent canonical transformation. The introduced concept "canonical transformation of the kinetic equation" has the following meaning: having received some adequate description of the kinetic equation in a Cartesian coordinate system, a transformation is carried out that allow representing the equation in a new Cartesian coordinate system in such a way that its structure corresponds to the canonical form. The basic postulate of chemical kinetics can be such a canonical type.

Half-a-century Burial of ρ, θ and φ in PDB

Since the launch of Protein Data Bank (PDB) in 1971, Cartesian coordinate system (CCS) has been the default approach to specify atomic positions in biomolecular experimental structures with X, Y and Z. In 2020, a local spherical coordinate system (LSCS) approach was proposed as an alternative to CCS, i.e., ρ, θ and φ. Recently, the continued application of deep learning technology in protein structure prediction (PSP) saw a leap forward in the accuracy of PSP, as evidenced by AlphaFold of Google’s DeepMind. However, there still is room for the improvement of the performances of PSP algorithms to date. Given that geometrically, CCS and LSCS are like the two sides of a coin, this short article puts forward a hypothesis that the time is now ripe to end the half-a-century burial of ρ, θ and φ in PDB, and use them as LSCS features for the design of novel PSP algorithms in future.

ДЕКАРТОВА СИСТЕМА КООРДИНАТ ЯК ІНСТРУМЕНТ ПОЗИЦІОНУВАННЯ ОБ’ЄДНАНИХ ТЕРИТОРІАЛЬНИХ ГРОМАД УКРАЇНИ

Problem statement. The need for positioning of amalgamated territorial communities (ATC) is due to their large number and lack of identification. Effective ATC positioning requires the development of appropriate tools to take advantage of the territory. The purpose of the article is to highlight the results of using the Cartesian system of coordination of the rear positioning of ATC in the context of their sustainable development, examples of ATC Luhansk region. The research object is ATC of Luhansk region. The methods used in the reaserch are: system, comparative and graphical analysis (the Cartesian coordinate system in space). The reaserch hypothesis was the preparation that the Cartesian coordinate system can be used as a tool for assessing the positions of ATCs in the context of their sustainable development. Presentation of the main material: on the basis of sustainable development estimates of ATC of Luhansk region their positions in the Cartesian coordinate system are determined with the help of geogebra software; to determine the best positions in the ATC of Luhansk region, it is proposed to use the vector of balanced positioning; it is established that the Lozno-Oleksandrivska ATC has the best balanced positioning.The originality and practical significance of the reaserch polarize the propositions of vicoristic Cartesian coordinate systems for the simplest tools for controlling the positions of ATC . Conclusions and prospects fot the further reaserches. For the assessment of ATC positions, it was used a middle ball of the development by three warehouse (economy, social, environmental). The result of the positioning of ATC of the Luhansk region was filled with a rosary behind an additional Cartesian coordinate system and software geogebra protection. The positions of ATC of the Luhansk region are displayed in the areas XOZ, XYZ, XOY. For setting the best positions at the ATC of the Luhansk region, the reference station is assigned to the vector of the balanced position, which dials up, for the most unbalanced positions from the Lozno-Oleksandrivska position. Subsequent reaserches may be directed to the value of visits for the purpose of the balanced development of ATC.

USING THE DUALITY PRINCIPLE WHEN CONSTRUCTING EXERCISES AT THE GEOMETRY LESSONS

Geometry is one of the complex disciplines where many facts are interconnected. It is possible to develop the idea of facts interrelations through correlation using the duality principle. The duality principle is known in projective geometry, mathematical logic. This principle is clearly pronounced in one of the theorems of new triangle geometry. The traditional analytical geometry course does not study the facts of new triangle geometry. To reinforce many topics of the analytical geometry course, for example, “The distance between two points”, “The symmetrical form of the equation of a line”, “The angle between two lines”, it is reasonable to consider some facts from the new triangle geometry in the Cartesian coordinate system. Thus, an element of novelty is introduced to the reviewed material. The guidebooks on triangle geometry solve tasks through classical approaches or applying barycentric coordinates not using analytical geometry formulas. The paper proposes the constructing technique for the couples of exercises using the duality principle in the plane geometry teaching methods. Tasks are constructed for the Cartesian coordinate system as this allows demonstrating the duality of points in the drawings. In the composed exercises, two drawings are constructed in parallel columns. In different cases, the points can be the triangle-apexes, an orthocenter, or a height base. The initial triangle sides are located on the axes of coordinates, and their side lengths set up Pythagorean triple for better understanding the task-solving algorithm by the students. The symmetrical form of the equation of a line shows the necessity of analytical study since it is difficult to check the distance from the orthocenter to the orthotriangle sides in the drawings due to the small value. For many such information units, the aggregation relationships (whole-part) are set up, reflecting the geometric embedding of components.

The effects of vertebral rotation on the position of the aorta relative to the spine in patients with adult degenerative scoliosis

Aims: This study aimed to explore the effects of vertebral rotation on the position of the aorta relative to the thracolumbar and lumbar spine, and to identify risk factors for vertebral rotation in patients with adult degenerative scoliosis (ADS). Methods: A total of 71 patients with ADS were divided into left scoliosis (LS) group ( n = 40 cases) and right scoliosis (RS) group ( n = 31cases) with well-matched demographics. Apical vertebrae, Cobb angle (°), coronal horizontal movement, thoracolumbar kyphosis (TLK) and Nash–Moe rotation classification were measured on X-ray. The Cartesian coordinate system was established on T2-MRI for each level of intervertebral disc on thracolumbar and lumbar spine, where aorta–vertebrae angle (α), aorta–vertebrae distance (d), and vertebral rotation angle (γ) for each level of T12-L1 to L3-L4 on MRI were defined within the Cartesian coordinate system. Results: There was no statistical difference in the distribution of apical vertebrae between LS and RS groups. Nash–Moe classification was of no significance between the two groups. When there was a larger Cobb angle and coronal horizontal movement, a greater γ in LS group and a lower γ in RS group were noted (both p < 0.001). There was no correlation among γ, α, and d in LS group ( p = 0.908 and 0.661, respectively) nor in RS group ( p = 0.738 and 0.289, respectively). In LS group, Nash–Moe classification correlated to Cobb angle, coronal movement and TLK. In RS group, it correlated to Cobb angle and coronal movement. Cobb angle was the risk factor for Nash–Moe classification in RS group while no factors were identified in LS group. Coronal movement was independent risk factor for γ ( p = 0.003) in LS group. Moreover, γ was affected by Cobb angle ( p = 0.001) and coronal horizontal movement ( p = 0.006) in RS group. Conclusion: Vertebral rotation could be predicted by Cobb angle or coronal horizontal movement measured on X-ray in ADS patients and aorta maintained in a relatively normal position in patients with ADS.

An improved model for performances calculation in spoke-type permanent-magnet machines considering magnetization orientation and finite soft-magnetic material permeability

Purpose In semi-analytical modeling of spoke-type permanent-magnet (PM) machines (STPMM), the saturation effect is usually neglected (i.e. iron parts are considered to be infinitely permeable) and the PM magnetization is assumed tangential (i.e. magnetization pattern is considered to be tangential-parallel). This paper aims to present an improved two-dimensional (2D) subdomain technique for STPMM with the PM magnetization orientation in quasi-Cartesian coordinates by using hyperbolic functions considering non-homogeneous Neumann boundary conditions (BCs) in non-periodic regions and by applying the interfaces conditions (ICs) in both directions (i.e. t- and θ edges ICs). Design/methodology/approach The polar coordinate system is transformed into a quasi-Cartesian coordinate system. The rotor and stator regions are divided into primary subdomains, and a partial differential equation (PDE) is assigned to each subdomain. In the PM region, the magnetization orientation is considered in the equations. By applying BCs, the general solution of the equations is determined, and by applying the ICs, the corresponding coefficients are determined. Findings Using the proposed coordinate system, the general solution of PDEs and their coefficients can mathematically be simplified. The magnetic field and non-intrinsic unbalanced magnetic forces (UMF) calculations have been performed for three different values of iron core relative permeability (200, 800 and ∞), as well as different magnetization orientation values (135 and 80 degrees). The semi-analytical model based on the subdomain technique is compared with those obtained by the 2D finite-element analysis (FEA). Results disclose that the PM magnetization angle can affect directly the performance characteristics of the STPMM. Originality/value A new model for prediction of electromagnetic performances in the STPMM takes into account magnetization direction, and soft magnetic material relative permeability in a pseudo-Cartesian coordinate system by using subdomain technique is presented.