Русский
09.2015 - Volume 19 Issue 3 (PDF)
The article presents a result on finding the minimum number \(f(n)\) of arithmetic progressions needed to obtain in union all natural numbers not divisible by n. Here n is an arbitrary natural number. In this case, two cases are investigated. In the first case, the progressions can intersect, in the second they cannot. In both cases, the authors of the article managed to find the exact value for the function \(f(n)\) and provide a constructive partition of this subset of the natural numbers into \(f(n)\) arithmetic progressions.
Keywords: Natural numbers, arithmetic progression, decomposition.
The authors introduce an extended superposition of automata. An extended superposition is a superposition of automata with a fixed addition of Boolean functions and a delay. The authors proved the algorithmic solvability of the expressibility problem for Medvedev group automaton functions, as well as linear automaton functions for the Medvedev group of automata, constant automata, linear automata.
Keywords: Automaton, superposition, algorithm.
The paper considers a model of a dynamic system consisting of a graph, each vertex of which is assigned the characteristics of "favorability" and "the number of elements in a given vertex". Each element tends to get to the most favorable vertex, while the presence of elements in a vertex is an unfavorable factor. It is shown that such a system always has a limit state, a class of systems is indicated for which this limit state depends only on the number of elements in the entire graph, but not on their distribution among the nodes
Keywords: dynamic systems, stabilization, automata modeling of migration processes.
This article considers the problem of selecting a fan according to specified parameters. Air flow and pressure will be used as the main parameters. For fans whose characteristics do not allow obtaining the required air flow and pressure, the problem of changing the characteristic by adjusting the operating frequency will be considered.
Keywords: PHP, industrial ventilation, frequency regulation.
This article considers the problem of calculating the characteristics of a water cooler. Given the input parameters for air (inlet air temperature, inlet air relative humidity, etc.), coolant (direct and return coolant temperature, impurity percentage, etc.) and the cooler itself (cross-sectional area, heat exchange area, number of circuits, distance between plates, etc.), it is necessary to calculate such values as outlet air temperature, outlet relative air humidity, total battery thermal power, condensate amount, battery hydraulic resistance and other important physical characteristics.
Keywords: PHP, industrial ventilation, heat exchangers.
The paper constructs a mathematical model of recommender systems. Estimates of the number of user types in the user matrix are provided. The NP-hardness of the problems of finding a representative and enumeration of user matrices is proved in the general case, and a polynomial algorithm for finding a representative of user matrices is given in the case where the number of user types is equal to two.
Keywords: recommender systems, matrix restoration, matrix replenishment, finding a representative of user matrices, enumeration of user matrices, estimating the number of user types, NP-hardness
An example of a closed class of automata with the operation of superposition, non-extensibility to a pre-complete class, is given.
Keywords: Automaton, superposition, pre-complete class.
In this paper, an undecidable superintuitionistic propositional calculus is constructed whose axioms contain only three variables.
Keywords:Superintuitionistic propositional calculus, undecidable calculus, Minsky machine.
The paper considers the problem of exact decoding of a linear Boolean function of arity n, which essentially depends on k variables. Exact values of the complexity of decoding for small k, and upper bounds for the general case are obtained.
Keywords: exact decoding of functions, linear Boolean functions.
The article proves the criterion of predictability of general regular super-events and the criterion of predictability of super-iterations of regular events. Algorithms for constructing predictive automata are given.
Keywords: event forecasting, automata, general regular events.
The article considers the class T of thin languages of regular languages with at most linear growth function. For these languages, the concept of two dimensions dim and Dim is introduced. A result is given on the values taken by these quantities on T. In addition, the set of all realizable pairs (dimP, DimP) is found, where on P ∈ T.
Keywords: spectrum, thin languages, dimension, growth function.
The author previously proved that pushdown automata preserve periodic sequences, and an upper bound for the maximum period length, exponential in the automaton characteristics, was given. Further, for the case when the alphabet of the store consists of one symbol, the author managed to lower the general estimate to a quadratic one. In this work, it is shown that in the case when the automaton store contains at least two symbols, it is impossible to significantly lower the upper estimate, since it was possible to construct examples of automata that generate sequences with an exponential period length.
Keywords: automaton with store memory, deterministic function, periodic sequences.
Countable criterion systems of closed subclasses in classes of linear-automaton functions over finite fields are obtained.
Keywords: finite automaton, linear-automaton function, composition operations, superposition operations, feedback, completeness problem, closed subclass, criterion system, adder, delay.