Four-vectors in relativity

“Four-vectors in relativity” gives a “soft” introduction to four-vectors by first setting up corresponding properties of three-vectors. These include the triangle rule for vector addition, and rotation of axes by a matrix multiplication. The physics of a three-dimensional system is unchanged by a rotation of the axes within which it is observed. Likewise the physics of a relativistic system is unchanged (“invariant”) under application of a Lorentz transformation.

Method of applied direction of mathematics in the aviation institution of higher education

The elements of the method of applied direction of mathematics in the aviation institution of higher education developed by us are considered. We use this technique in higher mathematics classes at the Flight Academy of the National Aviation University. Examples of a combination of fundamental mathematical concepts and practical methods of their application are given. We illustrate in detail the coordinate systems used in aviation. Among these systems are mobile and fixed coordinate systems. Problems of higher mathematics related to one or another coordinate system are indicated. For example, to record vector equations of motion in projections, moving coordinate systems are used, the beginning of which is located in the center of mass of the aircraft. Therefore, the study of the topic "Reflection of linear (vector) spaces" acquires a professional orientation. In particular, we present the formulas of coordinate transformations for parallel transfer and rotation of the axes. Note that the transformations of rectangular coordinate systems are used in aviation. Having considered aviation coordinate systems, the teacher is interested in students in the study of equations of motion, determination of accelerations, velocities and displacements. Methodical methods of formation of practical skills and abilities of future aviation specialists contribute to the implementation of the applied direction of mathematics. We have given some examples of methods of applied direction of mathematics in aviation in the sources that are currently published. The prospect of our research is to further improve practical approaches to solving problems of mathematical training of students of aviation institutions of higher education. This should help increase the level of methodological training of scientific and pedagogical staff of higher education institutions. At the same time, it should contribute to the improvement of methods of teaching mathematics in terms of its application. As a result, the graduate of the aviation institution of higher education must be ready for successful professional activity. Key words: applied direction of mathematics, coordinate systems in aviation, parallel transfer, rotation of axes, fundamental mathematical concepts in aviation.

Selected Concrete Models Studied Using Willam’s Test

Willam’s test is a quick numerical benchmark in tension–shear regime, which can be used to verify inelastic (quasi-brittle) material models at the point level. Its sequence consists of two separate steps: uniaxial tension accompanied with contraction—until the tensile strength is attained; and next for softening (cracking) of the material—tension in two directions together with shear. A rotation of axes of principal strains and principal stresses is provoked in the second stage. That kind of process occurs during the analysis of real concrete structures, so a correct response of the material model at the point level is needed. Some familiar concrete models are selected to perform Willam’s test in the paper: concrete damaged plasticity and concrete smeared cracking—distributed in the commercial ABAQUS software, scalar damage with coupling to plasticity and isotropic damage—both implemented in the FEAP package. After a brief review of the theory, computations for each model are discussed. Passing or failing Willam’s test by the above models is concluded based on their results, indicating restrictions of their use for finite element computations of concrete structures with predominant mixed-mode fracture.