Том 30, № 2 (2019)

We propose a coherent algorithm for the translation of hierarchical timed automata into networks of (planar) timed automata. This kind of translation is called flattening. The two types of timed automata are used in formal verification of real-time systems: systems of parallel interacting components whose execution essentially depends not only on the order of the events in the system, but also on the real time of these events. The concept of hierarchical timed automaton covers the syntactic variations that are used in existing studies and are non-comparable by their expressive power. The number of states in a flattened network of time automata is of the least order among the flattening results of existing studies.

We construct an infinite family of closed classes of partial monotone many-valued logic functions that include the class of everywhere defined functions monotone with respect to partially ordered sets of a special type.

The article proposes a synthesis method for amplifying networks of functional elements (ANFE) that establishes the asymptotic behavior of the Shannon function for the ANFE generalized depth, i.e., the depth of the “worst” Boolean function of n given variables, in a special basis (the depth model) where the element depth is determined both by its type and by its fan-out in the network. The asymptotic behavior of the Shannon function is established apart from a term logarithmic in n.

Wavefront reconstruction from local slope estimates is performed by a family of variational methods with a fractional order stabilizer designed to ensure best uniform reconstruction of the front on the entire frequency spectrum. Analytical and numerical investigation of the space-frequency characteristic of the family and numerical experiments with reconstruction of regular and turbulent wavefronts have demonstrated the advantage of the proposed method over the common Simpson method.

The article proposes a mathematical model linking GDP growth with changes in national debt. The model is based on a system of two linear ordinary differential equations. The rate of GDP growth is defined in the model as the difference between aggregate revenues and aggregate expenses, including debt service expenses. Possible types of GDP and national debt dynamics are investigated as a function of parameters. External investments increasing national debt may accelerate economic growth, whereas without new external loans GDP either does not grow or grows slowly. An equation describing the variation of relative external debt has been derived and fully investigated. It is shown that the external debt is never fully repaid, which is consistent with other models. Conditions of stable economic growth are derived, when GDP grows faster than or at the same rate as national debt, whereas the relative debt approaches a constant value. We investigate conditions that preclude a Ponzi game, so that a country cannot use new external debt to build a financial pyramid to repay old debt. The model features are demonstrated in application to statistical data from a number of countries. The model parameters are determined and growth trajectories are calculated for GDP and national debt.