By Steinitz' Theorem all triangulations of a sphere are generated from one triangulation with four vertices by certain sequences of operations called vertex splittings. A theorem of Barnette asserts that all triangulations of the projective plane can be generated from two irreducible triangulations.In the present work we obtain an analogous result for the torus: we show that all triangulations of the torus are generated by 21 irreducible triangulations (they are found explicitly) by applying the same vertex splitting operations. Two tables, one figure. Bibliography: 8 titles.The present work is the author's undergraduate thesis, written under the direction of Docent I. Kh. Sabitov at Moscow State University (1984).

A graph is called marked if its vertices differ one from the other by some label. As in [2], for a given marked graph G we denote by r(G, M 2) the number of all the pairwise different triangular packings (triangulations) Ti(G) of graph G on a connected, two-dimensional manifold (2-manifold) M2; moreover, two triangulations Til(G) and Ti2(G) with graph G will be

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The article proposes a method for predicting the daily energy consumption level for every day of a whole year, taking into account the season-al factor, based on only twelve actual power consumption data by the months of the year. Then a mathematical model is developed for moni-toring and controlling the level of electricity consumption on a daily basis, taking into account the seasonal factor. The model is consistent with a common model for the length of daylight (in hours). In addition, on the basis of this model, a method of monitoring and diagnostics of electricity consumption is presented, which will allow users to monitor the level of power consumption and be timely notified of any deviations from the theoretical level. Finally, this method gives rise to the operational principle for a proposed device, a smart energy meter, for detecting suspicious deviations from the theoretical level. The device will help timely detect over-consumption (or under-consumption) of electricity in order to take preventive measures. The proposed method consists of the following steps: (1) choice of a function to model the level of electricity consumption (theoretical calculated level), (2) choice of a tubular control neighborhood of the graph of the model function, (3) choice of a criterion on when the smart energy meter should notify the user of an unexpected deviation from the theoretical level in the case of exit from the tubular control neighborhood.