The specifics of the Galois field GF(257) and its use for digital signal processing.

An algorithm of digital logarithm calculation for the Galois field G F ( 257 ) is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it is advisable to use the specifics of quasi-Mersenne prime numbers, representable in the form p = 2 n + 1 , which includes the number 257. For fields G F ( 2 n + 1 ) , an alternating encoding can be used, in which non-zero elements of the field are displayed through binary characters corresponding to the numbers + 1 and - 1. In such an encoding, multiplying a field element by 2 is reduced to a quasi-cyclic permutation of binary symbols (the permuted symbol changes sign). Proposed approach makes it possible to significantly simplify the design of computing devices for calculation of digital logarithm and multiplication of numbers modulo 257. A concrete scheme of a device for digital logarithm calculation in this field is presented. It is also shown that this circuit can be equipped with a universal adder modulo an arbitrary number, which makes it possible to implement any operations in the field under consideration. It is shown that proposed digital algorithm can also be used to reduce 256-valued logic operations to algebraic form. It is shown that the proposed approach is of significant interest for the development of UAV on-board computers operating as part of a group.

The Method of Direct and Reverse Phase Portraits as a Tool for Systematizing the Results of Studies of Phase Transitions in Solutions of Thermosensitive Polymers.

It is shown that a more than significant amount of experimental data obtained in the field of studying systems based on thermosensitive hydrophilic polymers and reflected in the literature over the past decades makes the issue of their systematization and classification relevant. This, in turn, makes relevant the question of choosing the appropriate classification criteria. It is shown that the basic classification feature can be the number of phase transition stages, which can vary from one to four or more depending on the nature of the temperature-sensitive system. In this work, the method of inverse phase portraits is proposed for the first time. It was intended, among other things, to identify the number of phase transition stages. Moreover, the accuracy of this method significantly exceeds the accuracy of the previously used method of direct phase portraits since, for the first time, the operation of numerical differentiation is replaced by the operation of numerical integration. A specific example of the application of the proposed method for the analysis of a previously studied temperature-sensitive system is presented. It is shown that this method also allows for a quantitative comparison between the results obtained by the differential calorimetry method and the turbidimetry method. Issues related to increasing the resolution of the method of direct phase portraits are discussed.

Effect of the Formation of Hydrophilic and Hydrophobic-Hydrophilic Associates on the Behavior of Copolymers of N-Vinylpyrrolidone with Methyl Acrylate in Aqueous Solutions.

It has been shown that there exist conditions under which thermosensitive copolymers of N-vinylpyrrolidone with methyl acrylate form hydrophobic-hydrophilic associations, which are unstable dynamic meshes, the bonds in which are continuously broken and created again, and the nature of the formation of such meshes depends significantly on the proportion of the hydrophobic component in the copolymer. It is shown that the interaction of the above copolymers with polyacrylic acid results in the formation of not only classical interpolymer complexes, but also hydrophilic interpolymer associates, which also represent unstable networks existing in a dynamic mode. In such meshes, the molecules of the above copolymers serve as a kind of cross-agent connecting the polyacid molecules. There are also conditions under which such meshes acquire a complex structure, since unstable bonds between macromolecular tangles of both the same and different types take part in their formation. It is shown that the transition from the formation of interpolymer complexes to the formation of hydrophilic interpolymer associates can occur, among other things, due to changes in the acidity or concentration of low-molecular salt in solution.

Improving the efficiency of using multivalued logic tools: application of algebraic rings.

It is shown that in order to increase the efficiency of using methods of abstract algebra in modern information technologies, it is important to establish an explicit connection between operations corresponding to various varieties of multivalued logics and algebraic operations. For multivalued logics, the number of variables in which is equal to a prime number, such a connection is naturally established through explicit algebraic expressions in Galois fields. It is possible to define an algebraic δ-function, which allows you to reduce any truth table to an algebraic expression, for the case when the number of values accepted by a multivalued logic variable is equal to an integer power of a prime number. In this paper, we show that the algebraic δ-function can also be defined for the case when the number of values taken by a multivalued logic variable is p - 1, where p is a prime number. This function also allows to reduce logical operations to algebraic expressions. Specific examples of the constructiveness of the proposed approach are presented, as well as electronic circuits that experimentally prove its adequacy.

Features of digital signal processing algorithms using Galois fields GF(2n+1).

An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1. For such numbers, it is possible to implement a multiplication algorithm modulo p, similar to the multiplication algorithm modulo the Mersenne number. It is shown that for such numbers a simple algorithm for digital logarithm calculations may be proposed. This algorithm allows, among other things, to reduce the multiplication operation modulo a prime number p = 2n+1 to an addition operation.

Formation of Hydrophobic-Hydrophilic Associates in the N-Vinylpyrrolidone and Vinyl Propyl Ether Copolymer Aqueous Solutions.

Utilizing turbidimetry data, an examination is conducted on the behavior of solutions containing N-vinylpyrrolidone and vinyl propyl ether copolymer within a temperature range coinciding with the occurrence of a phase transition. The investigation reveals that within specific conditions prevailing in this domain, the emergence of entities denoted as hydrophobic-hydrophilic associates is conceivable. These entities are characterized by the presence of a relatively dense core, upheld by hydrophobic interplays, and they are proficient in effectively dispersing irradiation within the optical spectrum. Encircling this core is a hydrophilic periphery that impedes the formation of insoluble precipitates. The development of such associates transpires when hydrophobic interactions have attained a discernible prominence, although they remain inadequate to counteract the forces that drive the expansion of macromolecular coils. Under these circumstances, the energetically favored course of action entails the constitution of a core for the aforementioned associates, involving discrete segments from diverse macromolecules. Notably, the introduction of an additional constituent (ethanol) to the solution, which selectively mitigates hydrophobic interactions, serves to stabilize the hydrophobic-hydrophilic associations.

Improving the efficiency of using multivalued logic tools.

Multivalued logics are becoming one of the most important tools of information technology. They are in great demand for creation of artificial intelligence systems that are close to human intelligence, since the functioning of the latter cannot be reduced to the operations of binary logic. At the same time, the problem of improving the efficiency of using the results of research in multivalued logics, as well as the problem of interpreting variables of multivalued logic, is acute. These problems create certain interdisciplinary barriers and make it difficult to implement the results of research in the field of multivalued logics in other fields of knowledge. It is shown that the problem of interpreting multivalued logic variables can be removed by establishing correspondence with fuzzy logic variables. Improving the efficiency of using of operations of multivalued logics and their variables can be provided by using their close connection to Galois fields. This connection, among other things, makes it possible to reduce any operations of multivalued logics, the number of variables in which is equal to a prime number, to algebraic functions whose arguments take values in Galois fields. This allows, among other things, to eliminate the very cumbersome constructions used in works on multivalued logic and make its apparatus convenient for use in related scientific disciplines in information technology. Direct verification of the adequacy of algorithms based on the use of Galois fields can be carried out by means of radio-electronic circuits, examples of which are presented in the present paper.