THE APPLICATION OF ISOMORPHIC MATRIX REPRESENTATIONS FOR MODELING THE PROTOCOL FOR THE FORMATION OF SECRET KEYS-PERMUTATIONS OF HUGE SIZES

A The article considers the peculiarities of the application of isomorphic matrix representations for modeling the protocol of matching secret keys-permutations of significant dimension. The situation is considered when for cryptographic transformations of blocks with a length of 256 * 256 bytes, presented in the form of a matrix of a black-and-white image, it is necessary to rearrange all bytes in accordance with the matrix keys. To generate a basic matrix key and the appearance of the components KeyA and KeyB in the format of two black and white images, a software module using engineering mathematical software Mathcad is proposed. Simulations are performed, for example, with sets of fixed matrix representations. The essence of the protocol of coordination of the main matrix of permutations by the parties is considered. Also shown are software modules in Mathcad for accelerated methods that display the procedure of iterative permutations in a permutation matrix isomorphic to the elevation of the permutation matrix to the desired degree with a certain side, corresponding to specific bits of bits or other code representations of selected random numbers. It is demonstrated that the parties receive new permutation matrices after the first step of the protocol, those sent to the other party, and the identical new permutation matrices received by the parties after the second step of the protocol, ie the secret permutation matrix. Similar qualitative cryptographic transformations have been confirmed using the proposed representations of the permutation matrix based on the results of modeling matrix affine-permutation ciphers and multi-step matrix affine-permutation ciphers for different cases when the components of affine transformations are first executed in different sequences. , and then permutation using the permutation matrix, or vice versa. The model experiments performed in the study demonstrated the adequacy of the functioning of the models proposed by the protocol and methods of generating a permutation matrix and demonstrated their advantages.

A Study of the Second-Kind Multivariate Pseudo-Chebyshev Functions of Fractional Degree

Here, in this paper, the second-kind multivariate pseudo-Chebyshev functions of fractional degree are introduced by using the Dunford–Taylor integral. As an application, the problem of finding matrix roots for a wide class of non-singular complex matrices has been considered. The principal value of the fixed matrix root is determined. In general, by changing the determinations of the numerical roots involved, we could find n r roots for the n-th root of an r × r matrix. The exceptional cases for which there are infinitely many roots, or no roots at all, are obviously excluded.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.

Mysteries of the Heavenly Spheres: Astrology and Magic

This chapter studies astrology. The astronomical model of the cosmos described in astrology, with the divinities of the heavens proceeding in systematic and observable paths as they preside over the world of mortals, becomes one of the most important and widespread models for understanding the cosmos in the Greco-Roman world, starting with the Hellenistic period and increasingly so during the Roman Empire. Indeed, some form of this cosmological model appears almost everywhere in the Greco-Roman world in this time period, ranging from the most basic identification of traditional Greek and Roman gods with the visible planets to the most sophisticated and complicated systems detailed in the astrological manuals or Gnostic theologies. The most outstanding features of astrology that distinguish it from other forms of divination are its extreme complexity and systematicity. All forms of divination operate with a fixed matrix of signs and a random element of chance, but astrology works with a vast array of celestial signs, and the element of chance is limited to the moment of birth. While some of the multitude of uses of astrological ideas and images do not appear to be marked as abnormal or extraordinary, others bear the familiar stamp of strangeness that marks the practice as extraordinary, beyond the bounds of normal—magical.

Quantitative indicators of economic reliability analysis press-granulator with a fixed matrix

Purpose. Determine the main quantitative indicators for the economic analysis of the reliability of the pellet mill with a fixed matrix. Methods. Methods of grouping, system analysis, synthesis, scientific generalizations and the method of argumentation were used. Results. It is indicated that the main quantitative indicators for the economic analysis of machine reliability are: the cost of measures to increase reliability, the economic effect of increasing reliability and the payback period of measures to increase reliability. Equations are given for determining the payback period of measures to increase reliability and operating costs with increasing reliability. The conditions of economic feasibility of work to increase the level of reliability are determined. Conclusions. The calculations show that the annual economic effect of increasing the reliability of the pelletizer is 175.82 thousand UAH/year. Reliability enhancement measures reduce operating costs by 18.7% and payback periods are 2.55 years. Keywords: livestock, pellet press, reliability, efficient use, economic feasibility.

Numerical modeling of heat transfer in the fixed-matrix regenerator working in the Electric Thermal Storage heating system

The study presents the concept of Electric Thermal Storage (ETS) central heating system. Thermal Energy Storage (TES) is carried out in the fixed-matrix regenerator. The energy conservation equations, determined for the discharge period of the regenerator operation, are implemented in MATLAB numerical procedures based on the Finite Volume Method (FVM). In the model pressure drops within the system are calculated, both for the airflow through the inner tubes, and between the tubes. The flow distribution calculations show that the assumption of even air flow distribution would not be justified. Subsequently, the values of heat transfer coefficients are determined for the four distinct heat transfer surfaces, for the variable axial coordinate z and during the time of the system operation. The use of two different criterion equations is considered, for determining the mean Nusselt number Num for fluid flow through the concentric annular duct, as well as for the local Nusselt number Nuz calculated for the fluid flow through a circular or non-circular ducts. The most appropriate approach is selected by comparing the calculation results with experimental data. Taking into account the relative error, RMSE, and MAPE values calculated, it may be concluded that the Taler correlation – for non-circular ducts – gives results closer to the experimental data obtained.