### Working paper

## On projections of smooth and nodal plane curves

The research is related to the problem of coherent evolution of a domain-specific language (DSL) in response to evolution of the application domain and users’ capabilities. We offer a solution of that problem based on a particular model-driven approach. We give the whole definition of DSL in terms of model-oriented approach. Such definition allows us to define the DSL development using the mechanism of consecutive, consistent transformations between DSM, DSL meta-model and DSL concrete syntax model. In our approach we call such transformations as projections.

The article is related to the problem of sustainable flexibility of a domain-specific language (DSL) in response to evolution of the application domain and users’ capabilities. We offer a solution of that problem based on a particular model-driven approach. We propose to create a DSL structure from the domain-semantic-model (DSM) through the so-called semantic projection mechanism. The semantic projection is an operation, which is conducted over DSM. The result of the projection is a fragment of DSM, which describes the semantic model of a particular DSL dialect. We suggest to apply a group of model-to-model (M2M) transformations for practical implementation of semantic projections and producing corresponding DSL artefacts. We demonstrate the application of the proposed approach by the example in railway allocation domain.

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak's classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak's theorem to the case of finite characteristic.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.