Pages: 565-583
Published: 30.11.2019
Abstract: A new problem is solved in the article: for a given diagonal matrix ?pp=diag(?1,…,?p), ?1>…>?p>0, ?1+…+?p =р, it is required to find the values ??of the elements of 2 model submatrices Zmq, Zmp of the matrix Zmn=[Zmq(Zmp,], consisting of m values ??of n z-variables, m = q + p, q?p ,. The set of z-variables is divided into 2 groups: the z-variables z1,…,z6, are combined in the 1st group, z7,…,z12 - in the 2nd. The resulting 2 model submatrices Zmq, Zmp must be calculated after separate orthonormal transformations - model matrices Aqp and Bpp,, 2 matrices Ump, Vmp values ??of bi-orthogonal redundancy-canonical variables (u- and v-variables):(1/m)UTU=Ipp,(1/m)VTV=Ipp,(1/m)UTV=?pp=diag (?1,…,?p), ?1>…>?p>0. The model matrices Aqp and Bpp must have the algebraic properties of orthonormal matrices: AAT =I qq, BBT=I pp ,ATA =I pp,BTB=Ipp. The model submatrix Zmq must be computed using the transformation using the Aqp matrix, and the model submatrix Zmp must be calculated using the Bpp matrix. Orthonormal matrices Aqp, Bpp from PM AIKP [2-3] provide bi-orthogonality of the matrices Ump, Vmp: (1/m)UTV=?pp=diag(?1,…,?p). The model matrices of the Inverse Problem to be solved are calculated by modeling the historical principle “lower classes do not want, upper circles cannot”. As a result of mathematical modeling of the subject area, 2 factors (crisis generators) with negative dynamics of their curves are identified (Figures 2 and 3): for the lower classes - “the number of peasants who rented or bought land” (“weight” is b41 = 0.3580), and for upper circles - “degree distribution (implementation) of the idea of ??liberalism (“ weight ”is a41 = -0.50000).
Key words: redundancy-canonical variable, serfdom.
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