Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.
In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.
Äîâåäåíî, ùî âñi âàðiàíòè ïðÿìîêóòíî¨â'ÿçêè ïîïàðíî içîìîðôíi. Äîñëiäaeåíî âàðiàíòè ïðÿìîêóòíî¨â'ÿçêè ç ïðè¹äíàíîþ îäèíèöåþ òà ç ïðè-¹äíàíèì íóëåì.Êëþ÷îâi ñëîâà: âàðiàíò, ñåíäâi÷ íàïiâãðóïà, ïðÿìîêóòíà â'ÿçêà. Äëÿ êîaeíî¨íàïiâãðóïè (S, •) òà äîâiëüíîãî ôiêñîâàíîãî åëåìåíòà a ∈ S ìîaeíà çàäàòè íîâó áiíàðíó àñîöiàòèâíó îïåðàöiþ * a íà ìíîaeèíi S x * a y = x • a • y äëÿ äîâiëüíèõ x, y ∈ S. Îïåðàöiþ * a íàçèâàþòü ñåíäâi÷-ìíîaeåííÿì, à íàïiâãðóïó (S, * a ) ñåíäâi÷-íàïiâãðóïîþ ÷è âàðiàíòîì. Âàðiàíòè íàïiâãðóï âèâ÷àþòü ðiçíi àâòîðè ùå ç 60-õ ðîêiâ äâàäöÿòîãî ñòîëiòòÿ.  [11] ðîçãëÿäàþòüñÿ âàðiàíòè íàïiâãðóï ïåðåòâîðåíü, ÿêi é íàäàëi äîñëiäaeóâàëèñÿ, íàïðèêëàä, ó [3]. Äîñëiäaeåííÿ âàðiàíòiâ îõîïëþ¹ ðiçíi êëàñè íàïiâãðóï (äèâ., íàïðèêëàä, [7], òà ãëàâó 13 iç [5]). Âàðiàíòè íàïiâãðóï ïðÿìîêóòíèõ ìàòðèöü ðîçãëÿíóòi ó [4]. Ó áàãàòüîõ ïðàöÿõ (äèâ., íàïðèêëàä, [6] òà [12]) âèâ÷àëè iíòåðàñîöiàòèâíîñòi ìîíî¨äiâ, ÿêi òiñíî ïîâ'ÿçàíi ç âàðiàíòàìè. Âàðiàíòè ðåãóëÿðíèõ íàïiâãðóï äîñëiäaeóâàëè ó [9] òà [10]. Äëÿ êîìóòàòèâíèõ â'ÿçîê ç íóëåì ó [2] âñòàíîâëåíî êðèòåðié içîìîðôíîñòi äâîõ âàðiàíòiâ i êëàñèôiêîâàíî âàðiàíòè äåÿêèõ êîíêðåòíèõ â'ÿçîê.Â'ÿçêîþ íàçèâà¹òüñÿ íàïiâãðóïà, âñi åëåìåíòè ÿê iäåìïîòåíòàìè. Áóäåìî íàçèâàòè íàïiâãðóïó S ïðÿìîêóòíîþ â'ÿçêîþ, ÿêùî xyx = x äëÿ äîâiëüíèõ x, y ∈ S. Î÷åâèäíî, ùî òàêà íàïiâãðóïà ¹ ðåãóëÿðíîþ.Ìè äîñëiäaeó¹ìî âàðiàíòè ïðÿìîêóòíèõ â'ÿçîê i ïðÿìîêóòíèõ â'ÿçîê ç ïðè¹äíàíîþ îäèíèöåþ òà íóëåì.Ç òåîðåìè 1.1.3 [8] âèïëèâà¹, ùî äëÿ äîâiëüíî¨ïðÿìîêóòíî¨â'ÿçêè S iñíóþòü íåïîðîaeíi ìíîaeèíè X i Y òàêi, ùî íàïiâãðóïà S içîìîðôíà íàïiâãðóïi, âèçíà÷åíié 2020 Mathematics Subject Classication: 20M10
In this paper we consider variants of the lattice of partitions of a finite set and study automorphism groups of this variants. We obtain irreducible generating sets for of the lattice of partitions of a finite set. We prove that the automorphism group of the variant of the lattice of partitions of a finite set is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the direct product of the wreaths products, such that depends on the type of the variant generating partition and the second is defined by the certain set of symmetric groups.
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