Geometric Reflections: Why Should They Not Be Just a Flip?

Using descriptions from popular textbooks, the authors share the importance of introducing the definition so students can make sense of reflections both on and off the coordinate grid.

Automated determination of internal points of the coordinate grid of the blasted rock mass

An automated method for determining the internal points of the coordinate grid of the blasted rock mass is described. It is based on the method of determining the nodal points of the coordinate grid of the blasted rock mass, which is based on taking into account the dependencies that connect the initial parameters of the blasting rock mass with the final location of the fixed points of the blasted rock mass. The determining factors are the specific height and width of the collapse, the coefficient of loosening of the rocks. The method of analysis of experimental and industrial mass explosions in quarries, methods of analytical geometry, matrix theory and linear algebra are used. For the first time in mining, an analytical method has been developed for determining the internal points of the coordinate grid of an exploded block. It includes the established functions of the movement of nodal points, components of the vectors of movement of nodal and internal points of the coordinate grid. The established dependencies allow one to determine the displacements of any point inside the coordinate grid of the blasted block from the initial coordinates of the nodal and internal points.

SIMULATION OF MULTI-STAGE COLD FORMING OF A THIN-WALLED VESSEL

The method of construction and attestation of a numerical model of cold stamping of thin-walled products made of anisotropic metal sheet for the design of technological operations is proposed. The relations of the associated law of plastic flow with the Barlat flow function and isotropic strain hardening are used. The method of design and processing the experiment is proposed for their identification. The forming limit curve is approximated numerically by the Marciniak - Kuczynўski method, and for its identification it is proposed to use a failure test under uniaxial tension and press equipment as an experimental. To do this, a coordinate grid is applied to a flat blank by laser engraving, whose distortions near the zones of strain localization and failure of the vessel give additional points of the forming limit curve. The constants of the Peng - Landel potential are found to describe the elasticity of a polyurethane die under large deformations using tests for free and constrained compression. All tests according to the method were performed for low-carbon sheet steel DC04EK 0.7 mm and SKU-PFL polyurethane. A numerical model of the process in the LS-DYNA package is designed using material models from its library. The calculations according to the model were confirmed by experiment, for which the main deformations were determined by the distorted coordinate grid on the workpiece after each operation at the control points. The calculation of the sequence of stages of stamping, drawing and bulging of the workpiece in the production of the vessel with and without intermediate annealing is performed and the dangerous zones and mechanisms of their formation are determined.

Ultrathin Acoustic Parity-Time Symmetric Metasurface Cloak

Invisibility or unhearability cloaks have been made possible by using metamaterials enabling light or sound to flow around obstacle without the trace of reflections or shadows. Metamaterials are known for being flexible building units that can mimic a host of unusual and extreme material responses, which are essential when engineering artificial material properties to realize a coordinate transforming cloak. Bending and stretching the coordinate grid in space require stringent material parameters; therefore, small inaccuracies and inevitable material losses become sources for unwanted scattering that are decremental to the desired effect. These obstacles further limit the possibility of achieving a robust concealment of sizeable objects from either radar or sonar detection. By using an elaborate arrangement of gain and lossy acoustic media respecting parity-time symmetry, we built a one-way unhearability cloak able to hide objects seven times larger than the acoustic wavelength. Generally speaking, our approach has no limits in terms of working frequency, shape, or size, specifically though we demonstrate how, in principle, an object of the size of a human can be hidden from audible sound.

THEORETICAL UNDERPINNING OF FIELDS MAPPING AUTOMATION FOR IMPROVEMENT OF THE WAY OF SOIL SAMPLIN

The study objective – is the improvement of sampling of the soil by automation of field mapping. High labour and time requirements at all stages of performance study is the shortcoming of existing ways of soil sampling. In particu-lar, field mapping is carried out by both ways either geodetic methods and equipment, or Aerial Survey Cartog-raphy, however in all cases the analysis of data and work allotted to make quads in the field is carried out manually. Manual subjective correction of field mapping results is a disadvantage. The soil sampling way offered allows au-tomating this process, i.e. calculation of coordinates of quads, its entry and sampling points for robotics taking into account a field contour form parameters and a coordinate grid positioning, concentration of sampling points, a traffic route of robotic samplers between these points. The area of the field is divided into quads of correct (square or rec-tangular) or wrong (with the bent borders) forms in regard to this method. These quads are at the same time, cells of the coordinate grid put on the map of the field. Its parameters (a step and an azimuth) are set at the beginning of mapping. Coordinates of sampling points of the soil, on each quad, and a trajectory of their location, are determined taking into account a form of the concrete quad by the offered dependences. The offered way of sampling of the soil envisages the use of a special device for its implementation – the robotic sampler taking into account (in the course of mapping) coordinates estimated prior to the work start and its performance automatically. Field mapping and co-ordinates calculation of all critical points is to be carried out on the basis of the developed mathematical depend-ences and technological schemes automatically, using navigation systems of GLONASS or GPS standards.

Matter in curved spacetime

This chapter is concerned with the laws of motion of matter—particles, fluids, or fields—in the presence of an external gravitational field. In accordance with the equivalence principle, this motion will be ‘free’. That is, it is constrained only by the geometry of the spacetime whose curvature represents the gravitation. The concepts of energy, momentum, and angular momentum follow from the invariance of the solutions of the equations of motion under spatio-temporal translations or rotations. The chapter shows how the action is transformed, no longer under a modification of the field configuration, but instead under a displacement or, in the ‘passive’ version, under a translation of the coordinate grid in the opposite direction.