№ 5 (2023)

The article discusses the various numerical functions that determine the degree of "similarity" of the two given final sequences. These similarity measures are based on the concept we define of embedding in a sequence. A special case of such an attachment is the usual sub-subsequence. Other cases further require equality of distances between adjacent sub-sequence symbols in both sequences. This is generalization of the concept of a sequence segment (substring) in which these distances are unit. In addition, equality of distances from the beginning of the sequences to the first embedding symbol or from the last embedding symbol to the end of the sequences may be required. Except these last two cases, the attachment can be in a sequence several times. The literature uses functions such as the number of common attachments or the number of attachment occurrence pairs in a sequence. In addition to them, we enter three more functions: the sum of the lengths of total investments, the sum of the minima of the number of occurrences of a common embedding in both sequences and the similarity function based on the largest number of symbols of the common embedding. In total, 20 numerical functions are considered, for 17 of which algorithms (including new ones) of polynomial complexity are proposed, for two more functions, algorithms have exponential complexity with reduced a measure of degree. The Conclusion gives a brief comparative description of these investments and functions.

The second part of the study on the topic of automated depersonalization of personal data is presented. The review and analysis of the prospects for research, performed earlier, is supplemented here by a practical result. A model of the depersonalization process is proposed, reducing task of ensuring anonymity of personal data to manipulation of samples of different types of random elements. Accordingly, the key idea of transforming data to ensure their anonymity, provided that utility is maintained, is to apply the synthesis method, i.e. complete replacement of all unpublished data with synthetic values. The proposed model identifies a set of element types for which synthesis patterns are proposed. The set of patterns compiles the depersonalization algorithm by the synthesis method. Methodically, each template is based on a typical statistical tool – frequency probability estimates, nuclear Rosenblatt-Parsen density estimates, statistical averages and covariances. The application of the algorithm is illustrated by a simple example from the field of civil air transportation.

Systems of linear ordinary differential equations with the coefficients in the form of infinite formal power series are considered. The series are represented in a truncated form, with the truncation degree being different for different coefficients. Induced recurrent systems and literal designations for unspecified coefficients of the series are used as a tool for studying such systems. An algorithm for constructing Laurent solutions of the system is proposed for the case where the determinant of the leading matrix of the induced system is not zero and does not contain literals. The series included in the solutions are still truncated. The algorithm finds the maximum possible number of terms of the series that are invariant with respect to any prolongations of the truncated coefficients of the original system. The implementation of the algorithm as a Maple procedure and examples of its usage are presented.

Mathematical formulation of Bohr’s complementarity principle leads to the concepts of mutually unbiased bases in Hilbert spaces and complementary quantum observables. In this paper, we consider algebraic structures associated with these concepts and their applications to constructive quantum mechanics. We also briefly discuss some computer-algebraic approaches to the problems under consideration and propose an algorithm for solving one of them.

We propose an efficiently verifiable lower bound on the rank of a sparse fully indecomposable square matrix that contains two non-zero entries in each row and each column. The rank of this matrix is equal to its order or differs from it by one. Bases of a special type are constructed in the spaces of quadratic forms in a fixed number of variables. The existence of these bases allows us to substantiate a heuristic algorithm for recognizing whether a given affine subspace passes through a vertex of a multidimensional unit cube. In the worst case, the algorithm may output a computation denial warning; however, for the general subspace of sufficiently small dimension, it correctly rejects the input. The algorithm is implemented in Python. The running time of its implementation is estimated in the process of testing.