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The Puma lineage within the family Felidae consists of 3 species that last shared a common ancestor around 4.9 million years ago. Whole-genome sequences of 2 species from the lineage were previously reported: the cheetah (Acinonyx jubatus) and the mountain lion (Puma concolor). The present report describes a whole-genome assembly of the remaining species, the jaguarundi (Puma yagouaroundi). We sequenced the genome of a male jaguarundi with 10X Genomics linked reads and assembled the whole-genome sequence. The assembled genome contains a series of scaffolds that reach the length of chromosome arms and is similar in scaffold contiguity to the genome assemblies of cheetah and puma, with a contig N50 = 100.2 kbp and a scaffold N50 = 49.27 Mbp. We assessed the assembled sequence of the jaguarundi genome using BUSCO, aligned reads of the sequenced individual and another published female jaguarundi to the assembled genome, annotated protein-coding genes, repeats, genomic variants and their effects with respect to the protein-coding genes, and analyzed differences of the 2 jaguarundis from the reference mitochondrial genome. The jaguarundi genome assembly and its annotation were compared in quality, variants, and features to the previously reported genome assemblies of puma and cheetah. Computational analyzes used in the study were implemented in transparent and reproducible way to allow their further reuse and modification.

When fighting against terrorism in modern armed conflicts, combat vehicles, including tanks, are widely used. To minimise own losses of vehicles and personnel for overthrowing enemy is a relevant task. To solve it, the paper considers certain events in the history of the Great Patriotic War, which are associated with battle of tanks that spring an ambush. A mathematical model of the battle is built. The state graph of the system is given. Using this graph, a probability of tank kills and a ratio of mathematical expectations of losses have been calculated. This mathematical model generalizes the models, previously published in this journal, based on the Markov chain apparatus. The paper gives an example of calculations for this model in the particular case in which experimental data are used as a basis. The ratios of mathematical expectations of losses of the warring parties are obtained. Further, we consider the mathematical models, in which it is assumed that probabilities for tank crews to provide operations of targets detection in firing are known. With technology development and its mathematical support it becomes increasingly more real. The formulas to obtain the probability of tank kills are given according to the graph of states using the known probabilities of transition from one state to another. In each of the three mathematical models under consideration there is a graph of the system state, which allows calculation of the tank kills probability. We have analysed the models to prove a significant dependence of the loss ratio of the warring parties on the number of firing positions used by the tank in ambush in case re-siting is unnoticeable for the enemy. The authors-considered models that use the examples of historical events confirm that the tactics of organising and conducting ambushes in tank battles can be successfully used nowadays, when the technology intensiveness of the opposing forces significantly grows. The obtained results can be applied to organise and conduct tank ambushes in modern armed conflicts and fight against terrorist army.

Evolutionary history and taxonomic position for cryptic species may be clarified by using molecular and cytogenetic methods. The subterranean rodent, the Alay mole voleEllobiusalaicusVorontsov et al., 1969 is one of three sibling species constituting the subgenus Ellobius Fischer, 1814, all of which lost the Y chromosome and obtained isomorphic XX sex chromosomes in both males and females.E.alaicusis evaluated by IUCN as a data deficient species because their distribution, biology, and genetics are almost unknown. We revealed specific karyotypic variability (2n = 52–48) inE.alaicusdue to different Robertsonian translocations (Rbs). Two variants of hybrids (2n = 53, different Rbs) withE.tancreiBlasius, 1884 were found at the Northern slopes of the Alay Ridge and in the Naryn district, Kyrgyzstan. We described the sudden change in chromosome numbers from 2n = 50 to 48 and specific karyotype structure for mole voles, which inhabit the entrance to the Alay Valley (Tajikistan), and revealed their affiliation asE.alaicusby cytochromeband fragments of nuclearXISTandRspo1genes sequencing. To date, it is possible to expand the range ofE.alaicusfrom the Alay Valley (South Kyrgyzstan) up to the Ferghana Ridge and the Naryn Basin, Tien Shan at the north-east and to the Pamir-Alay Mountains (Tajikistan) at the west. The closeness ofE.tancreiandE.alaicusis supported, whereas specific chromosome and molecular changes, as well as geographic distribution, verified the species status forE.alaicus. The case ofEllobiusspecies accented an unevenness in rates of chromosome and nucleotide changes along with morphological similarity, which is emblematic for cryptic species.

The taxonomy of the genus Calomyscus remains controversial. According to the latest systematics the genus includes eight species with great karyotypic variation. Here, we studied karyotypes of 14 Calomyscus individuals from different regions of Iran and Turkmenistan using a new set of chromosome painting probes from a Calomyscus sp. male (2n = 46, XY; Shahr-e-Kord-Soreshjan-Cheshme Maiak Province). We showed the retention of large syntenic blocks in karyotypes of individuals with identical chromosome numbers. The only rearrangement (fusion 2/21) differentiated Calomyscus elburzensis, Calomyscus mystax mystax, and Calomyscus sp. from Isfahan Province with 2n = 44 from karyotypes of C. bailwardi, Calomyscus sp. from Shahr-e-Kord, Chahar Mahal and Bakhtiari-Aloni, and Khuzestan-Izeh Provinces with 2n = 46. The individuals from Shahdad tunnel, Kerman Province with 2n = 51–52 demonstrated non-centric fissions of chromosomes 4, 5, and 6 of the 46-chromosomal form with the formation of separate small acrocentrics. A heteromorphic pair of chromosomes in a specimen with 2n = 51 resulted from a fusion of two autosomes. C-banding and chromomycin A3-DAPI staining after G-banding showed extensive heterochromatin variation between individuals.

The genus status of Urocricetus was defined recently based on morphological and molecular data. Even though the amount of evidence for a separate phylogenetic position of this genus among Cricetinae continues to increase, there is still no consensus on its relationship to other groups. Here we give the first comprehensive description of the U. kamensis karyotype (2n = 30, NFa = 50) including results of comparative cytogenetic analysis and detailed examination of its phylogenetic position by means of numerous molecular markers. The molecular data strongly indicated that Urocricetus is a distant sister group to Phodopus. Comparative cytogenetic data showed significant reorganization of the U. kamensis karyotype compared to karyotypes of all other hamsters investigated earlier. The totality of findings undoubtedly means that Urocricetus belongs to a separate divergent lineage of Cricetinae.

The emergence of new intellectual (or smart) technologies changes not only the mentality of human society, but also all spheres of its life. This change is associated with the emergence and development of a powerful and expressive language—the language of numbers and digital technology. At present, intelligent technologies and intelligent systems and smart systems are becoming an integral phenomenon in almost all spheres of human life. This, in turn, requires new courses and new mathematical methods in the field of smart systems theory. The paper contains a description of such a course developed by the authors in the field of algebraic methods in the digitalization of economic smart systems and in the field of education. The course “Algebraic methods in digitalization of smart systems” was developed by the authors on the basis of works on algebraic formalization and identification of smart systems carried out in the period from 1998 to the present time. Almost simultaneously with the development, fragments of the course were tested at a seminar on financial mathematics at the Academy of Budget and Treasury of the Ministry of Finance of the Russian Federation.

This paper discusses the issue of algebraic methods in the digitalization of smart systems that allow building complex characteristics of qualitative and quantitative measures of the system, and on this basis, a comprehensive unified indicator of the system’s some properties is introduced. Within the framework of the Zermelo–Fraenkel set theory axiomatic, the existence of such an indicator is proved. In fact, thereby the solution of the generalization of multicriteria Operation Research problem has been obtained within the framework of the Zermelo–Fraenkel set theory axiomatic. Received results can be used in e-learning and in smart learning theory in assessment of control function of smart learning system’s functioning.

In this chapter, we consider smart systems whose models of determining factors are solvable groups. Also, quasi-fractal systems which modeled by quasi-fractal groups of determining factors, the levels of which are solvable groups and simple groups are considered. In Chap. 9, we have considered sustainability issues in relation to systems whose models are quasi-fractal algebraic systems. In this chapter: we shall introduce and consider the concept of structural sustainability of a closed associative system with a feedback, we shall show that closed innovation systems with commuting determining factors do not have the property of structural stability, a closed associative system S with a feedback modeled by a free non-abelian group is structurally sustainable with a minimum regulation range, theorems about tensor estimates of structurally sustainable groups.

Recently the concepts of smart systems, smart education, smart engineering, smart medicine, smart city, smart universities, smart technology and smart communication were introduced in scientific methodology and practice for a modern society. All of these entities are the components of a new system—smart society system. These components must be equipped with corresponding connections that ensure their reliability, sustainability and security, or, in other words, to have an appropriate smart infrastructure. It turns out that we approach the concept of smart infrastructure from the standpoint of tensor estimation. So, the main questions to consider here are the following ones

In this chapter the following basic questions are considered: models’ presentation forms, statistics methods of system identification, algebraic methods of systems identification, transition from the element-wise model of the system to the model of factors determining the system, fractal forms of model representation and systems identification methods, quasi-fractal algebraic systems

In this chapter the following basic questions are considered: Cayley diagrams, Cayley complexes and Cayley groups graphs, the relationship between them through the basic concepts of category theory, Van Kampen diagrams, relationship of Cayley diagrams and Van Kampen diagrams, a brief overview of graph study methods, probabilistic properties of graphs, the use of first-order logic in the study of graphs, category concept, examples, role in mathematics and use in systems theory, the relationship between groups and graphs through category theory, the possibility of transferring the theory of \( P \)—purities of algebraic systems to graphs, probability-isomorphic finite quasi-fractal groups as analogues of giant components.

In this chapter the following basic questions are considered: complex systems’ problems of measuring, generalizations of classical scales. Multidimensional, matrix and lattice scales, examples. coding as a tool to measure students’ level of knowledge, quasi-fractals and synergistic effects. Scales fixing the synergistic effect and the time (level) series determined by them, measurement scales for quasi-fractal algebraic systems. quasi-fractal scale measurability level.

In this chapter the following basic questions are considered: natural fractal algebraic systems through series of examples from mathematics, natural sciences, telecommunications, the concept of quasi-fractal homomorphism and target quasi-fractal of a system, system target functions, regulation of a quasi-fractal system. Regulatory functions of the system. Theorems explaining the appearance of mutations.

In the first Chapter, the problem of identification of a smart system is considered from the standpoint of compliance of the functioning of the system with its purpose of functioning and evaluation of this compliance. The main issues here are: In the first Chapter, the problem of identification of a smart system is considered from the standpoint of compliance of the functioning of the system with its purpose of functioning and evaluation of this compliance. The problem of identification of smart systems is to answer the question: does the system satisfy the objective of its operation at each moment in time, and if so, how much? Thus, the problem of identification is divided into the following two tasks

In this chapter the following basic question is considered: The test system is a measurement system through assessments of students’ knowledge. The main issues here are: the question of assessing the adequacy of the results of measuring the real level of knowledge and skills of students, the question of a comprehensive assessment of the level of assimilation of the system of knowledge that connects quantitative and qualitative indicators. The main issues here are the question of assessing the adequacy of the measurement results to the real level of knowledge, skills of students. In this regard we shall consider two models of the algebraic formalization of the representation of the knowledge system—in the form of a free group of factors that determine the knowledge system, and in the form of a semantic network, which we represent using finite Boolean algebra. These models allow us to show that all binary tests i.e. tests involving answers in the form of either “yes” or “no”, can be solved true, not knowing the specific nature of the proposed questions. Then we shall show that probability measure can be used as a measure of the level of assimilation of the knowledge system represented by the semantic network.

In this chapter, we introduce the concept of parametric algebraic potential in line with the algebraic formalization of the system.

The mandarin vole, Lasiopodomys mandarinus, is one of the most intriguing species among mammals with non-XX/XY sex chromosome system. It combines polymorphism in diploid chromosome numbers, variation in the morphology of autosomes, heteromorphism of X chromosomes, and several sex chromosome systems the origin of which remains unexplained. Here we elucidate the sex determination system in Lasiopodomys mandarinus vinogradovi using extensive karyotyping, crossbreeding experiments, molecular cytogenetic methods, and single chromosome DNA sequencing. Among 205 karyotyped voles, one male and three female combinations of sex chromosomes were revealed. The chromosome segregation pattern and karyomorph-related reproductive performances suggested an aberrant sex determination with almost half of the females carrying neo-X/neo-Y combination. The comparative chromosome painting strongly supported this proposition and revealed the mandarin vole sex chromosome systems originated due to at least two de novo autosomal translocations onto the ancestral X chromosome. The polymorphism in autosome 2 was not related to sex chromosome variability and was proved to result from pericentric inversions. Sequencing of microdissection derived of sex chromosomes allowed the determination of the coordinates for syntenic regions but did not reveal any Y-specific sequences. Several possible sex determination mechanisms as well as interpopulation karyological differences are discussed.

The history of each human chromosome can be studied through comparative cytogenetic approaches in mammals which permit the identification of human chromosomal homologies and rearrangements between species. Comparative banding, chromosome painting, Bacterial Artificial Chromosome (BAC) mapping and genome data permit researchers to formulate hypotheses about ancestral chromosome forms. Human chromosome 13 has been previously shown to be conserved as a single syntenic element in the Ancestral Primate Karyotype, in this context, in order to study and verify the conservation of primate chromosomes homologous to human chromosome 13, we mapped a selected set of BAC probes in three platyrrhine species, characterised by a high level of rearrangements, using fluorescence in situ hybridisation (FISH). Our mapping data on Saguinus oedipus, Callithrix argentata and Alouatta belzebul provide insight into synteny of human chromosome 13 evolution in a comparative perspective among primate species, showing rearrangements across taxa. Furthermore, in a wider perspective, we have revised previous cytogenomic literature data on chromosome 13 evolution in eutherian mammals, showing a complex origin of the eutherian mammal ancestral karyotype which has still not been completely clarified. Moreover, we analysed biomedical aspects (the OMIM and Mitelman databases) regarding human chromosome 13, showing that this autosome is characterised by a certain level of plasticity that has been implicated in many human cancers and diseases.

One of the problematic issues of education is a comparative analysis of the results achieved by students in the learning process. The results may vary, for example, in breadth, in depth of knowledge, different areas of mathematics can be mastered by the same student at completely different levels, etc. The proposed smart algebraic approach that uses a probabilistic approach, semantic networks, and marked graphs for the analysis of learning outcomes allows you to measure learning outcomes, taking into account the structure of the knowledge system, rank of students according to their level of knowledge, evaluate their strengths and weaknesses, identify gaps among students in knowledge, and create recommendations on the teaching methodology. Smart algebraic model of learning outcomes analysis uses algebraic formalization of systems and probabilistic estimation methods.

A new approach to ensuring operational reliability is proposed. In this approach, three conditions for ensuring operational reliability must be satisfied. A mathematical apparatus for calculating the reliability of such structures is developed.

The taxonomy of the genus Calomyscus remains controversial. According to the latest systematics the genus includes eight species with great karyotypic variation. Here, we studied karyotypes of 14 Calomyscus individuals from different regions of Iran and Turkmenistan using a new set of chromosome painting probes from a Calomyscus sp. male (2n = 46, XY, Shahr-e-Kord-Soreshjan-Cheshme Maiak Province). We showed the retention of large syntenic blocks in karyotypes of individuals with identical chromosome numbers. The only rearrangement (fusion 2/21) differentiated Calomyscus elburzensis, Calomyscus mystax mystax, and Calomyscus sp. from Isfahan Province with 2n = 44 from karyotypes of C. bailwardi, Calomyscus sp. from Shahr-e-Kord, Chahar Mahal and Bakhtiari-Aloni, and Khuzestan-Izeh Provinces with 2n = 46. The individuals from Shahdad tunnel, Kerman Province with 2n = 51–52 demonstrated non-centric fissions of chromosomes 4, 5, and 6 of the 46-chromosomal form with the formation of separate small acrocentrics. A heteromorphic pair of chromosomes in a specimen with 2n = 51 resulted from a fusion of two autosomes. C-banding and chromomycin A3-DAPI staining after G-banding showed extensive heterochromatin variation between individuals.

Sokolov's dwarf hamster (Cricetulus sokolovi) is the least studied representative of the striped hamsters (Cricetulus barabensis species group), the taxonomy of which remains controversial. The species was described based on chromosome morphology, but neither the details of the karyotype nor the phylogenetic relationships with other Cricetulus are known. In the present study, the karyotype of C. sokolovi was examined using cross-species chromosome painting. Molecular and cytogenetic data were employed to determine the phylogenetic position of Sokolov's hamster and to analyze the potential pathways of chromosome evolution in Cricetulus. Both the chromosome and molecular data support the species status of Sokolov's hamster. Phylogenetic analysis of the CYTB data placed C. sokolovi as sister to all other striped hamsters (sequence divergence of 8.1%). FISH data revealed that the karyotype of C. sokolovi is highly rearranged, with the most parsimonious scenario of its origin implying at least 4 robertsonian events and a centromere shift. Comparative cytogenetic data on Cricetinae suggest that their evolutionary history includes both periods of chromosomal conservatism and episodes of rapid chromosomal change.

The involvement of chromosome changes in the initial steps of speciation is controversial. Here we examine diversification trends within the mole voles Ellobius, a group of subterranean rodents. The first description of their chromosome variability was published almost 40 years ago. Studying the G-band structure of chromosomes in numerous individuals revealed subsequent homologous, step-by-step, Robertsonian translocations, which changed diploid numbers from 54 to 30. Here we used a molecular cytogenetic strategy which demonstrates that chromosomal translocations are not always homologous, consequently, karyotypes with the same diploid number can carry different combinations of metacentrics. We further showed that at least three chromosomal forms with 2n = 34 and distinct metacentrics inhabit the Pamir-Alay mountains. Each of these forms independently hybridized with E. tancrei, 2n = 54, forming separate hybrid zones. The chromosomal variations correlate slightly with geographic barriers. Additionally, we confirmed that the emergence of partial or monobrachial homology appeared to be a strong barrier for hybridization in nature, in contradistinction to experiments which we reported earlier. We discuss the possibility of whole arm reciprocal translocations for mole voles. Our findings suggest that chromosomal translocations lead to diversification and speciation.

Recently the concepts of smart systems, smart education, smart engineering, smart medicine, smart city, smart universities, smart technology and smart communication were introduced in scientific methodology and practice for a modern society. All of these entities are the components of a new system -- smart society system. These components must be equipped with corresponding connections that ensure their reliability, or, in other words, to have an appropriate smart infrastructure. One of the most important issues in this regard is a feedback and a control in smart system. This paper describes our approach to an application of statistical methods to automated knowledge control. The obtained research outcomes clearly show that if testing is used to test knowledge level, then a) it gives adequate outcomes of knowledge assessment only under repeated regular use of testing procedure and b) the final assessment should be obtained using the averaging procedure.

The test system is a measurement system through assessments of students’ knowledge. The main issues here are (a) the question of assessing the adequacy of the results of measuring the real level of knowledge and skills of students and (b) the question of a comprehensive assessment of the level of assimilation of the system of knowledge that connects quantitative and qualitative indicators. Both of these questions are related to the main problem—the problem of smart system identification. To answer the first question, we consider two models of algebraic formalization of the knowledge system’s representation—in the form of a free group of factors determining the knowledge system, and in the form of a semantic network, which we represent using finite Boolean algebras. Both models show that all binary tests, i.e., tests involving answers in the form of either “yes” or “no”, can be solved correctly, without knowing the specific essence of the proposed questions. In regards to the second question, we proposed a tensor estimate of the effectiveness of the functioning of a smart system with sufficiently rigid restrictions. Such an assessment can be determined, in particular, over a field of two elements for evaluating binary tests. In this paper, we show that this assessment can be viewed as coding the current state of the e-learning system, which is important for e-learning in relation to the issue of protection and storage of information. Similar algorithms are available for evaluating multi-choice tests. Algebraic smart systems’ formalization technique is used.

Optimal design of IPsec-based virtual private networks (VPN), in general, depends on multiple factors and parame- ters, such as the VPN's architectural model, hardware and software setups and the technical platform solutions, network topo- logy models, modes of the tunnel's operation, levels of the Open Systems Interconnection (OSI) model, encryption/decryption algorithms, modes of cipher operation, security protocols, security associations and key management techniques, connectivity modes, parameters of security algorithms, computer architectures, the number of tunnels in the VPN, and other factors. This paper presents an innovative approach to using methods of linear programming with risks to solve a multi objective optimiza- tion problem of VPN design. In particular, it describes the proposed conceptual VPN design model, VPN information security space, index of effectiveness for multi objective optimization, a new classification of scales, a system-based approach to risks and mathematical modeling of a risk, a hierarchy of scripts and a theorem of description of passwords, VPN design optimization process and particular procedures, sets of legal and illegal VPN design methods, and a VPN design optimization algorithm based on multi objective optimization by linear programming with risks models. The proposed and developed VPN design optimization algorithm was tested by developing specific VPN design methods for various types of VPN users.

Researchers have assembled thousands of eukaryotic genomes using Illumina reads, but traditional mate‐pair libraries cannot span all repetitive elements, resulting in highly fragmented assemblies. However, both chromosome conformation capture techniques, such as Hi‐C and Dovetail Genomics Chicago libraries and long‐read sequencing, such as Pacific Biosciences and Oxford Nanopore, help span and resolve repetitive regions and therefore improve genome assemblies. One important livestock species of arid regions that does not have a high‐quality contiguous reference genome is the dromedary (Camelus dromedarius). Draft genomes exist but are highly fragmented, and a high‐quality reference genome is needed to understand adaptation to desert environments and artificial selection during domestication. Dromedaries are among the last livestock species to have been domesticated, and together with wild and domestic Bactrian camels, they are the only representatives of the Camelini tribe, which highlights their evolutionary significance. Here we describe our efforts to improve the North African dromedary genome. We used Chicago and Hi‐C sequencing libraries from Dovetail Genomics to resolve the order of previously assembled contigs, producing almost chromosome‐level scaffolds. Remaining gaps were filled with Pacific Biosciences long reads, and then scaffolds were comparatively mapped to chromosomes. Long reads added 99.32 Mbp to the total length of the new assembly. Dovetail Chicago and Hi‐C libraries increased the longest scaffold over 12‐fold, from 9.71 Mbp to 124.99 Mbp and the scaffold N50 over 50‐fold, from 1.48 Mbp to 75.02 Mbp. We demonstrate that Illumina de novo assemblies can be substantially upgraded by combining chromosome conformation capture and long‐read sequencing.

One of the most important questions in the development of the work of the system of Smart Education is the problem of quality control of its functioning, and therefore the questions of system connections. Links in Smart Education are characterized by different types of assessments, such as exam grades, matriculation certificate obtained at the end of school, ranking assessments, etc. So, to analyze links one needs to use method that can combine qualitative and quantitative characteristics. We propose to study connections of Smart Education System using the method of algebraic formalization. This method makes it possible to describe the hierarchy of system’s structural links, to set off the levels of the system’s links, and to prove theorem on the description of the system’s links of each finite level. Different types of system’s links are singled out. The notion of a factor - fractal system on the links of finite levels are defined. The classification of the system’s links based on the number of synergetic effects of the group of the system’s links of a finite level is constructed. There are several practice applications of this method in the paper. Thus, the justification of Granovetter hypothesis on the connections in social systems has been obtained. The next example relates to Smart Education. The structure of ranking systems for evaluating the effectiveness of the functioning of Smart University is proposed based on the formalization of the process of decomposition of system. Tensor efficiency index is introduced.

In this chapter we consider the following main questions: different approaches to the definition of duality in Smart Systems Theory, measurement of the system’s links strength, the group of links on the Cayley Graph of the system, the concept of efficiency and its formalization, the concept of \(P\)-efficiency of a system, \(P\)-subgroups of effective links of a system. In addition to Chap. 4 in this chapter we shall continue to study the links of the system and define another group of system’s links as a group defined on the Cayley graph of the group of factors that determine the system. Then we proceed to study the subgroups of effective connections of the system. We shall also introduce the notion of a common efficiency problem with risks. The next question that we shall consider in this chapter is the use of duality in systems theory. We shall give a brief survey of some of the methods that are useful to investigate the duality.

In this chapter we shall continue to study the concept of efficiency, and in line with this concept we introduce the definition of an innovative smart system. After that in terms of algebraic formalization we shall describe the structure of innovative smart systems. Then we shall consider the concept of pseudo-innovation system, dual to the concept of an innovative smart system. Also an algebraic formalization of some properties of innovative smart systems and pseudo-innovative systems is constructed. Some examples of the use of these concepts consumed in the expert systems in training, and in the economy are given. After that we shall continue the empirical study of the process of system’s decomposition using the example of the decomposition of the education system on the basis of these questions. The algorithm for a comprehensive assessment of the effectiveness of the functioning of the innovation system based on the tensor estimation of system performance is also proposed in this chapter. It is proposed to use homomorphisms of the group G s of factors defining the system S into the group GL(n,∆) of linear homogeneous transformations of the vector space R n as tensor estimates of the efficiency of the functioning of the system S. One can also consider homomorphisms of the group of factors G s that define a system S in the group GL (n,∆) of linear homogeneous transformations of the vector space ∆n over an arbitrary field ∆.

Communications among the systems factors are characterized by the interactions between the systems factors. In our case these are a binary operation of composition of the factors and an unary operation \({\square}^{ - 1}\), which characterized the feedback of the system. So Cayley table [1], of the group \(\varvec{G}_{S}\) of factors describing the system S plays an important role in describing the system’s connections. In this chapter it is from this position we shall begin to consider the question of the sustainability of the system which is defined in fact by the internal structure of its connections, robust and interchangeability of structural resources. There after we shall continue to study the sustainability of the system in the Chap. 10 from the position of the usage of numerical characteristics associated with Cayley table of the group of factors determining the system. Some useful ideas can be find in [2, 3].

In Serdyukova (Algebra & Logic 30:432–456, 1991, [1]), on the basis of a study Yu. L. Ershov’s works about profinite groups, a method has been proposed for isolating and studying purities or pure embeddings in a special class of algebraic systems that is the class of groups, which allowed to generalize the known results of the theory of purities of abelian groups to the case of arbitrary non-abelian groups (purities by predicates). In Serdyukova (Optimization of tax system of Russia, parts I and II, 2002, [3]), Serdyukova and Serdyukov (The new scheme of a formalization of an expert system in teaching, 2014, [4]), Serdyukova et al. on the basis of system approach. Springer, Berlin, pp 371–380, 2015 [5]) and Serdyukova and Serdyukov (Modeling, simulations and optimization based on algebraic formalization of the system, pp 576–582, 2015, [6]) a method for modeling the final states of the system and determining the number of final states using the technique of group theory has been developed. An important question when studying the properties of a system and the process of its functioning, and in particular when studying the properties of a smart system and its functioning, is the question of how to determine that a system or a smart system ceases to satisfy some property P or some complex of properties \(\varPi\) . To answer this question, we introduce the notion of a partial probability measure on the set of unary predicates defined on the class of groups and closed with respect to taking subgroups and factor-groups.

In this chapter before we begin to examine pro-P-groups, we return again to the special case in which the factors affecting the system determine the group. In this case the system is a closed associative system with a feedback. Here we dwell briefly upon the modeling of “identical” factors with respect to the structure that act on the system. The question arises as to how all possible structures of connections between factors acting on the system can be described. We shall use the automorphism group of the group of factors that determine the system to this purpose. After that we recall the definition and basic information about algebraically compact groups that are necessary for the study of innovative and pseudo-innovative systems. Algebraically compact groups are in some way a generalization of divisible groups in two following directions: the first line (1) is distinguished as a direct summand from the group containing it, when (2) certain conditions are imposed on how the subgroup is contained in the overgroup. If a divisible group can be defined as a group distinguished as a direct summand from any group that contains it, then an algebraically compact group is a group distinguished as a direct summand from any group that contains it as a pure subgroup.

We begin this chapter from the consideration the basic properties that determine the system: integrity, internal and external attributive features that determine the system, that is, distinguishing precisely this system from all others, in the language of algebraic formalization. Then the integrity property is generalized to the case of P-integrity and P-internal and P-external attributive characteristics of the system, allowing to classify the properties of the system according to their various components. The notions of a quasi-stable system with respect to the property P and the innovation system with respect to the property P are introduced. While constructing the algebraic formalization of smart systems, we shall take into account the Godel incompleteness theorem, the essence of which is that it is impossible to describe the system by using the means of this system only. As well a hierarchy of different levels links of the system is constructed. A theorem on the description of the system’s links is proved. In addition, on the base of the theory of binary relations constructed by A. I. Mal’tsev, a classification of the binary relations of a system of each finite level is upbuild. The obtained results are applied to construct smart system’s axiomatic and to models describing the system’s synergistic effects and the processes of system’s decomposition and synthesis.