scholarly journals XVII. On a new geometry of space

1865 ◽  
Vol 155 ◽  
pp. 725-791 ◽  
Keyword(s):  
Analytical Geometry ◽  
Infinite Space ◽  
Linear Complexes

I. On Linear Complexes of Right Lines . 1. Infinite space may be considered either as consisting of points or transversed by planes. The points, in the first conception, are determined by their coordinates, by x, y, z for instance, taken in the ordinary signification; the planes, in the second conception, are determined in an analogous way by their coordinates, introduced by myself into analytical geometry, by t, u, v for instance. The equation tx + uy + vz + 1 = 0 represents, in regarding x, y, z as variable and t, u, v as constant, a plane by means of its points. The three constants t, u, v are the coordinates of this plane. The same equation, in regarding t, u, v as variable, x, y, z as constant, represents a point by means of planes passing through it. The three constants are the coordinates of the point.

I. On a new geometry of space

1865 ◽  
Vol 14 ◽  
pp. 53-58 ◽  
Keyword(s):  
Analytical Geometry ◽  
Infinite Space

Infinite space may be considered either as consisting of points or transversed by planes. The points, in the first conception, are determined by their coordinates, by x, y, z for instance, taken in the ordinary signification; the planes, in the second conception, are determined in an analogous way by their coordinates, introduced by myself into analytical geometry, by t, y v for instance. The equation tx + uy + vz = 1 represents, in regarding x, y, z as variable, t, u , v as constant, a plane by means of its points. The three constants t, u, v are the coordinates of this plane. The same equation, in regarding t, u, v as variable, x, y, z as constant, represents a point by means of planes passing through it. The three constants x, y, z are the coordinates of this point.

scholarly journals Influence of Copper(I) Halides on the Reactivity of Aliphatic Carbodiimides

2020 ◽  
Vol 3 (1) ◽  
pp. 20
Author(s):  
Valentina Ferraro ◽  
Marco Bortoluzzi
Keyword(s):  
Density Functional Theory ◽  
Electronic Structure ◽  
General Formula ◽  
Electron Density ◽  
Density Functional ◽  
Functional Theory ◽  
Fukui Functions ◽  
Bond Lengths ◽  
Linear Complexes ◽  
Nitrogen Bond

The influence of copper(I) halides CuX (X = Cl, Br, I) on the electronic structure of N,N′-diisopropylcarbodiimide (DICDI) and N,N′-dicyclohexylcarbodiimide (DCC) was investigated by means of computational DFT (density functional theory) methods. The coordination of the considered carbodiimides occurs by one of the nitrogen atoms, with the formation of linear complexes having a general formula of [CuX(carbodiimide)]. Besides varying the carbon–nitrogen bond lengths, the thermodynamically favourable interaction with Cu(I) reduces the electron density on the carbodiimides and alters the energies of the (NCN)-centred, unoccupied orbitals. A small dependence of these effects on the choice of the halide was observable. The computed Fukui functions suggested negligible interaction of Cu(I) with incoming nucleophiles, and the reactivity of carbodiimides was altered by coordination mainly because of the increased electrophilicity of the {NCN} fragments.

scholarly journals On the numerical values of the roots of the equation cosx = x

1890 ◽  
Vol 9 ◽  
pp. 80-83
Author(s):  
T. H. Miller
Keyword(s):  
Analytical Geometry

The angles which satisfy the equationx=cosx,occur in the solution of certain problems in analytical geometry.It is easy to show that one value of the angle is coscoscos…cos 1, when the cosine is taken successively to infinity.

Surface Texture Measurement with Profile Method Using Six-Axis Coordinate Measuring Machine

2021 ◽  
Vol 410 ◽  
pp. 872-877
Author(s):  
Andrey V. Kochetkov ◽  
Andrey A. Troshin ◽  
Oleg V. Zakharov
Keyword(s):  
Surface Texture ◽  
Measurement Errors ◽  
Coordinate Measuring Machine ◽  
Texture Measurement ◽  
Analytical Geometry ◽  
Complex Shapes ◽  
Measurement Results ◽  
Measuring Machine ◽  
Cartesian Coordinate ◽  
Filtering Effect

Currently the measurement of surface texture in mechanical engineering is traditionally carried out using profilometers. Modern profilometers do not allow measuring of surfaces with complex shapes. This is due to the different sensitivity of the sensor and the discreteness of the movements along the axes of the Cartesian coordinate system. Coordinate Measuring Machines are devoid of such a drawback. However, stylus of the coordinate measuring machine has a diameter many times larger than the diamond stylus of the profilometer. Therefore, there is a mechanical filtering effect, that affects the results of measuring the parameters of the surface texture. In this paper a mathematical model of the contact of the spherical stylus and a rough surface based on analytical geometry is proposed. Influence of the diameter of the spherical stylus on the maximum measurement errors of a amplitude parameters are investigated. Seven amplitude parameters Rp, Rv, Rz, Ra, Rq, Rsk, Rku of the surface texture are modeled. Coordinate measuring machine and profilometer with stylus diameter of 2 μm measurement results are compared. it was concluded that the stylus diameter of the coordinate measuring machine when measuring the surface texture should be no more than 20 μm.

Way to improve efficiency of teaching linear algebra to students with health disabilities by means of enlarged didactic units

2021 ◽  
pp. 65-71
Author(s):  
A.N. Semakin ◽  
◽  
G.P. Emgusheva
Keyword(s):  
Students With Disabilities ◽  
Linear Algebra ◽  
Teaching Methods ◽  
Educational Material ◽  
Educational Process ◽  
Teaching Mathematics ◽  
Abstract Concepts ◽  
State Technical ◽  
Analytical Geometry ◽  
Time Gap

Presented is enlarged didactic units developed by P.M. Erdniev for teaching mathematics, that improves the ability of students to understand educational material. This effect is achieved as a result of organizing the educational process according to the principles of complementarity of teaching methods and the spatial and temporal combination of interrelated elements of knowledge. Among the basic mathematical disciplines taught at Bauman Moscow State Technical University for students with disabilities the most difficult discipline is Linear Algebra, which is due to its content and various problems with students’ health. Linear Algebra is largely based on the theory from Analytical Geometry that is quite well perceived by students with disabilities. A comparative analysis of these disciplines shows that the enlarged didactic units linking these disciplines together can significantly simplify the study of Linear Algebra. However, due to difficulties with scheduling the time gap between these disciplines can be up to two semesters, which makes it impossible to directly use the enlarged didactic units. To solve this problem, we introduce the supporting discipline "Cognitive technologies for supporting the discipline Linear Algebra" which runs in parallel with Linear Algebra and which is based on the material of Analytical Geometry. Forming its content in close connection with Linear Algebra in accordance with the main principles of the enlarged didactic units, we smoothly lead students to the understanding of abstract concepts of Linear Algebra, reducing the complexity of learning.

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