2019 year, volume 23, issue 2 (PDF)

Botkholov A.J. Recovery of two notes sounding simultaneously

In this work recovery of two notes sounding simultaneously in terms of amplitudes and frequencies is considered. The methods of recovery of amplitudes of two notes which were played at the same time by two different musical instruments are given. Cases of existence and not existence of two solutions for a problem of recovery of notes are described.

Keywords: basic tone, overtone, amplitude, frequency, bending around note, total spectrogram.

Kushnareva L., Kuzminykh D. Persistent homology of Markov chains and its application to natural language processing

In this work we introduce new persistent topological invariants for Markov chains. We also demonstrate the way of using these invariants for natural language processing task.

Keywords: Topology data analysis, persistent homology, Markov chains, natural language processing.

Kazakov I.B. Reliabilty criterion for channels with prohibitions

We investigate the possibility of reliable transmission in a situation when an adversary can prohibit some characters, and a set of prohibitions can change at every clock cycle. We show that reliable transmission is possible if and only if the cardinality of the alphabet n and the number of allowed characters k satisfy the inequality n ≤ 2k−2.

Keywords: covert channels, walks in a plane, character prohibition, transmittable language.

V.N. Kozlov Image segmentation and shape-preserving transformations

The image in the paper is a finite (non-empty) set of points in Euclidean spaces of different dimensions. Briefly, the work consists in having an image belonging to a certain image to extract from this image what could be called a description of this image.

Keywords:mathematical definition of the image, image recognition.

Piven N.A. Some properties of permutation construction for parametric assignment of quasigroups

We analyse so–called permutation construction, introduced before. It turns out, that it’s redundant, so we propose an improvement, which makes it injective for linear functions and possibly injective without conditions.

Keywords: Quasigroup, Latin square, parametric assignment, proper families of functions.

Sobyanin P.I. About algorithm which checks if subquasigroup exists in a given quasigroup

Algorithm which checks if subquasigroup exists in a given quasigroup. Performance of its consequent and CUDA parallel versions are compared.

Keywords: quasigroup, subquasigroup, GPU, CUDA.

Vedernikov I.K. Criterion for almost complete prediction of a superword in a multi-valued alphabet

The machine predicts the next character of the input sequence if it outputs that character the moment before. The present paper considers whether an arbitrary superword in multivalued alphabet can be almost completely predicted or not. The paper provides a theorem that enables to restrict the class of machines, with the help of which superwords are predicted. Moreover, the paper presents the criterion for almost complete predicting.

Keywords: almost complete predicting, predicting machine, prediction of superwords by a machine, criterion for predicting.

Efimov A.A. The upper estimate of the volumetric power consumption of the circuits that implement boolean operators.

In this work volume schemes which are generalization of plane schemes in space are considered. The class of the schemes implementing boolean operators was considered. For this class upper assessment of potential — a measure of the power equal to quantity of the circuit elements giving unit on this input pattern is received. It is shown that any operator of n variables can be realized with a volume scheme whose potential does not exceed \(O(m \cdot 2^{n/3})\) if m ≤ n and \(O(\frac{m}{n} \cdot \sqrt[3]{n} \cdot 2^{n/3})\) if m > n.

Keywords: schemes from functional elements, volume schemes, scheme power, potential.

Kan A. N. Questions of expressibility in the class of matched functions

In this paper we consider the 2-completeness in the class of matched functions P. This class was considered earlier in [3, 4]. It is precomplete in the class PL of piecewise-linear functions. There are two precomplete classes: the class of continuous fucnton, the class of matched finite function.

Keywords: Class of piecewise-linear functions, the class of piecewise-linear continuous functions, the class of matched functions, class of finite functions, the class of mathced finite function, 2-precompleteness, the Heaviside function.

Kapustin I.S. On the elementary expressibility in predicate logic

New mathematics concepts are often introduced with some quantifier definitions. If we have a sufficiently large stock of such notions, it can allow to reformulate the new quantifier definitions in a quantifier-free form. This makes the problem of finding basic concepts, which make further quantifiable definition redundant, worth considering. Creating computer programs that automatically introduce such bases is also worth considering. In this paper we observe 3 simple cases of reducing the quantifier expressions to the quantifier-free ones. We investigate predicates and functions defined by ∈ predicate on the set \(\textit{Z} \cup\ 2^Z\), where Z is the set of integers. We consider predicates expressed by ∈ predicate on the set of points of the plane and the lines lying in it. Finally, predicates expressed on the set of natural numbers by the | predicate on it are also considered. Bases were found in all 3 cases.

Keywords: predicate logic, quantifier definitions

Konovalov A. Yu. The Zermelo–Fraenkel set theory is not sound with respect to the constructive sematics based on hyperarithmetic sorts

A semantics of the realizability based on hyperarithmetical sorts for formulas of the language of set theory is introduced. The soundness of axioms of the Zermelo–Fraenkel set theory with respect to this semantics is studied.

Keywords: constructive semantics, realizability, axiomatic set theory, hyperarithmetical sorts.

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