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Том 30, № 1 (2019)
- Год: 2019
- Статей: 10
- URL: https://journals.rcsi.science/1046-283X/issue/view/15454
I. Discrete Models of Information Systems
Universal Functions of Two Variables for Special Values of k
Аннотация
We consider the notion of universal function such that a subset of the function’s values defines any function from some set. For the set of linear functions, we consider all the combinations of the number of variables and the number of values, except four-valued functions of two variables.
1-6
Article
Finding Maximal Independent Elements of Products of Partial Orders (The Case of Chains)
Аннотация
We consider one of the central intractable problems of logical data analysis – finding maximal independent elements of partial orders (dualization of a product of partial orders). An important particular case is considered with each order a chain. If each chain is of cardinality 2, the problem involves the construction of a reduced disjunctive normal form of a monotone Boolean function defined by a conjunctive normal form. An asymptotic expression is obtained for a typical number of maximal independent elements of products for a large number of finite chains. The derivation of such asymptotic bounds is a technically complex problem, but it is necessary for the proof of existence of asymptotically optimal algorithms for the monotone dualization problem and the generalization of this problem to chains of higher cardinality. An asymptotically optimal algorithm is described for the problem of dualization of a product of finite chains.
7-12
Circuit Complexity of k-Valued Logic Functions in One Infinite Basis
Аннотация
We investigate the realization complexity of k -valued logic functions k 2 by combinational circuits in an infinite basis that includes the negation of the Lukasiewicz function, i.e., the function k−1−x, and all monotone functions. Complexity is understood as the total number of circuit elements. For an arbitrary function f, we establish lower and upper complexity bounds that differ by at most by 2 and have the form 2 log (d(f) + 1) + o(1), where d(f) is the maximum number of times the function f switches from larger to smaller value (the maximum is taken over all increasing chains of variable tuples). For all sufficiently large n, we find the exact value of the Shannon function for the realization complexity of the most complex function of n variables.
13-25
High-Accuracy Bounds of the Shannon Function for Formula Complexity in Bases with Direct and Iterative Variables
Аннотация
We consider the realization of Boolean functions by formulas with restrictions on superpositions of basis functions such that superposition is allowed only by iterative variables. For a number of special symmetrical bases, we establish new high-accuracy bounds of the Shannon function L(n) for the complexity of realization of Boolean functions dependent on n direct variables.
26-35
Single Fault Diagnostic Tests for Inversion Faults of Circuit Elements Over Some Bases
Аннотация
The article shows that, in a number of bases, every Boolean function can be realized by an irredundant combinational circuit that admits a single fault diagnostic test of length not greater than 4 with respect to inversion faults on the element outputs.
36-47
48-54
Analytical Solution of the Quantum Master Equation for the Jaynes–Cummings Model
Аннотация
The article considers the dynamics of resonator photon occupancy (the probability of finding a photon in the resonator), atom excitation, and sink photon occupancy when a single photon escapes into the sink from an optical resonator populated by an atom in the ground and excited states. The photon-occupancy dynamics of the cavity is investigated using the Lindblad equation.
68-79
A Three-Dimensional Deconvolution Algorithm Using Graphic Processors
Аннотация
An iterative algorithm is described for three-dimensional deconvolution in the Fourier plane using parallel computations on CPU and GPU. The algorithm demonstrates easy scalability and can process any number of input images of any size. It is only limited by the local storage volume.
80-90
Mathematical Modeling of the Dynamics of Plasma Heating in a Magnetic Tube During Solar Flares
Аннотация
A mathematical model describing the initial phase of flare heating in the solar corona is proposed. The model is based on the nonlinear heat equation with a sign-changing volume source, which is obtained by reduction of the stationary- plasma energy equation. Flares are assumed to arise as a result of sausage-type instabilities in magnetic tubes and formation of collapsing magnetic traps. A source function is chosen, and model parameters are fitted. Calculations are performed, and the formation of thermal structures under supercritical perturbations against a homogeneous temperature background is studied. It is shown that, during the flare, structures are created in which the energy release half-width shrinks over time. The decrease of the emission measure observed in the early phase of the flare is associated with the decrease of the flare filling factor due to the decrease of structure half-widths.
91-97
II. Mathematical Modeling
Numerical Analysis of the Integral Equation Method for the Computation of the Electromagnetic Field in a Nonhomogeneous Medium
Аннотация
The article considers mathematical modeling of the electromagnetic field in a nonhomogeneous medium by the integral equation method. The case of high-contrast conducting media is studied in detail, with the conducting nonhomogeneity embedded in a poorly conducting medium. The analysis of the integral equation, in this case, has shown that the solution deteriorates when the conducting nonhomogeneity is inside a low-conductivity layer. It is shown that this effect can be overcome by Dmitriev’s method of elevated background conductivity. The contrast effect is most pronounced for the H -polarized two-dimensional electromagnetic field in a nonhomogeneous medium. The numerical experiment has accordingly been conducted for this particular case. The solution computed by the integral equation method with elevated background conductivity is compared with the solution computed by the finite-difference method. The results of the two methods show excellent fit.
55-67