A certain non-linear differential equation containing a power of unknown function being the solution is considered with application to selected geotechnical problems. The equation can be derived to a linear differential equation by a proper substitution and properties of operations G and S.

In the paper there has been made an advantage of the non-classical operational calculus to determination of the response of the certain discrete time-systems. The Z-transform is often used to analysis of the stationary discrete time-systems. However, the use of the Z-transform to determination of the response especially of the non-stationary discrete time-systems is doubtful or may cause complications. This method leads to differential equations of n-th order of variable coefficients, whose solutions are very difficult or impossible. The non-classical operational calculus can be used to analysis both of the stationary and non-stationary discrete time-systems. The presented method with the use of the Heaviside operator soon leads to the target without unnecessary differential equations.

The authors of this paper examine the ancient concepts of translatio, imitatio and aemulatio. The text goes over some problems of the heritage of antiquity and its reception in European culture of the early modern period. These questions were discusssed during the international conference “Heredes et scrutatores. Attitudes towards Antiquity in the Renaissance and in the Early Modern Period”, which was held on 19–20 May 2016 at the University of Warsaw. It celebrated the 200th anniversary of classical studies at this university. The conference seeked to explore the changing attitudes towards the heritage of classical antiquity in post-classical European culture. The scholars participating in the meeting tried to (re)examine the diversity of these attitudes in the period between the Late Middle Ages and Early Modern Times and to reflect on a number of related problems, among which were the theoretical viewpoints that had been suggested to describe this diversity. One of them, which gave its name to this conference, distinguishes between two general approaches: that of the “users”, concentrated on adapting the classical legacy by means of procedures inherited from the ancient Romans, and that of the “researchers”, which replaced the former procedures with ones typical of scholarly cognition. The participants discussed theoretical issues and concrete cases illustrating the ways that the intellectuals of the Renaissance and Early Modern Times approached the Greek and the Roman legacy. The connections between past and present attitudes towards antiquity have also been be the subject of the debate.

In this work, safety analysis at the railway level crossings is presented using advanced mathematical modelling. Resistivity of track subgrade panels is taken into account. The analysis does not refer to the assessment of the current regulations. Specific cases of generalized dynamic system are considered by introducing operations S=Δ, S=P defined in space C(N) of real sequences. In this model, generalized discrete exponential and trigonometric functions that reflect the oscillatory nature of the analysed quantities are used. The advantage of the analyzes is the avoidance of numerical errors. We show also the importance of the resistivity of track subgrade panels in safety at the level crossings. The safety at the level crossings can be increased through providing track subgrade panels with appropriate resistivity to minimize negative effect of stray currents. The results may be used to evaluate selected safety indicators as well as to predict safety levels and to determine the ways of improving safety.