Abstract: As part of an interdisciplinary approach, historians are increasingly using the ideas and methods of the mathematical sciences in their research. This, for example, arises in reconstruction problems from the field of historical geography, when the initial data is insufficient for traditional analysis. The subject of this study is to model the system of spatial distribution of the medieval population and demonstrate its results using a specific example. To take into account the condition of territory heterogeneity during modeling, the author develops a variational approach proposed by S.M. Huseyn-Zade. Using the methods of the calculus of variations and Pareto optimality within the framework of this approach makes it possible to identify and numerically describe the relationships between the optimal parameters of the system. To solve this problem of reconstruction, a model of a two-level hierarchy of placement of centers was previously proposed. Its approbation on the material of the scribal description of the Shelonskaya Pyatina of the Novgorod land at the end of the 15th century showed a high degree of correlation between real and theoretical data. The first surviving scribal description of the Pskov lands dates back to the second half of the 80s XVI century. It records a huge mass of abandoned lands as a result of the Livonian War. To assess the scale of this crisis, we first need data on the size of the rural population in the previous period of time. Unfortunately, the scribal description of the mid-16th century, which is closest to it in time, has not survived. However, the above-mentioned first surviving scribal description contains retrospective information about an earlier time, which makes it possible to apply the modeling results to solve the desired reconstruction problem, which is novel.

Keywords: linear regression, functional, calculus of variations, Pareto equilibrium, hierarchy of centers, population density function, pogosts, center placement system, scribe 's book, mathematical modeling

Abstract: The aim of the study is to reconstruct the demographic strategies of rural societies. The object of the study is individual settlements and village councils (groups of settlements) of the Tambov and Tver regions (a total of 2861 settlements and 371 village councils). The presented work is undertaken within the framework of a large project on fractal modeling of demographic strategies of the agrarian population of European Russia in a long historical retrospective – since the middle of the XIX century. The chronological framework of the article covers not only the agricultural crisis of the 1990s, but also a certain recovery of some rural settlements in the 2000s - 2010s. To conduct experiments with the proposed computer model, a database was created in which the main parameters of the studied settlements and village councils are presented in a formalized form. The authors conclude that, despite all the differences, the demographic strategies of Tver and Tambov settlements evolve within the same pattern: Tver and Tambov regions are simply in slightly different phases of the same process. Peasant society in the Tver region has less potential as a migration donor and a base for natural growth. The older and northern Tver Region has been following the path of de-settlement and urbanization for a longer time. Tambov peasant society is somewhat less depleted and, therefore, could demonstrate great demographic success in the case of a successful demographic policy.

Keywords: death rate, birth rate, migration, Tver region, Tambov region, rural population, demographic strategies, historical demography, fractal modeling, demographic factors

Abstract: The subject of this study is the modeling of the spatial distribution of the medieval population. To take into account the conditions of heterogeneity of the territory in modeling, the author develops a variational approach proposed by S.M. Huseyn-Zadeh. The model is based on the idea of settlement, placement of the population on the territory as a process in which each of its participants is guided in their activities by achieving goals dictated by natural considerations, for example, maximum accessibility to their cultivated area, maximum proximity to the central settlement (center). Using the methods of calculus of variations and the concept of Pareto optimality within this approach makes it possible to identify and numerically describe the relationships between the optimal parameters of the system. In the previous paper, the author proposed a model of a one-level hierarchy of the placement of centers (central settlements). Additional analysis of historical material leads to the need to refine the model, namely, adding a hierarchy of centers to it. In this regard, this paper proposes a model of spatial placement with a two-level hierarchy of centers. The approbation of the proposed model is carried out on the material of the scribal books of the Shelonskaya Pyatina of the Novgorod land of the end of the XV century. The high degree of conformity of the obtained theoretical and empirical data allows us to consider mathematical modeling as an adequate and convenient auxiliary tool in studying the nature and dynamics of historical rural settlement systems. The possibilities of the proposed modeling also in terms of filling data gaps are demonstrated by the example of the task of reconstructing the approximate population of the Shelonskaya Pyatina of the considered time period.

Keywords: hierarchy of centers, pogosts-places, linear regression, functional, Pareto equilibrium, population density function, calculus of variations, center placement system, scribe 's book, mathematical modeling

Abstract: One of the actual problems of Byzantine studies is the estimation of the size of the Genoese trading community of Constantinople, which played a critical role in the fate of late Byzantium. To solve this problem the historian A.L. Ponomarev proposed to use mathematical methods based on data from indirect sources - notarial deeds preserved in the State Archives of Genoa. These deeds were drawn up to fix commercial transactions, agreements on the creation of commercial partnerships, the hiring of ships, wills, the purchase and sale of houses, goods and people. In addition to the obligatory mention in the deed form of the names of the contracting parties and witnesses to the transaction, it may also, depending on its type, contain the names of guardians, recipients of the will and other third parties. Thus, these data on the clientele of Genoese notaries represent a rather impressive and valuable array of information, which may indirectly indicate the size of the entire trading Genoese community of Byzantium. To solve this problem, the author of this paper draws on the ideas and methods of the theory of random placements, which is an intensively developing area of mathematical statistics. It is based on constructing a linear estimate of the value and assumes a random sample. The result obtained is compared with the estimate from the previous paper by the author, which is based on another method of mathematical statistics and is quite close to the value of A.L. Ponomarev - 688 people.

Keywords: Constantinople, Genoese, notarial deeds, frequency of occurrence, ranged series, statistical estimation, sample, size of a finite population, linear estimate, random placement

Abstract: In the process of studying the systems of historical settlement, researchers are increasingly drawing on ideas and methods of mathematical modeling. The most popular - the probabilistic approach - is based on the idea that a random factor plays a predominant role in the process of spatial distribution of the population. In particular, according to the widely used Poisson model, events involving the emergence of new (and the disappearance of old) settlements on disjoint sites and time intervals are independent of each other, and the probability of two or more such events at one site is extremely small (the corresponding random variable is distributed according to Poisson's law). The purpose of this article is to familiarize the reader with the basic traditional principles of this approach, with the features of its methodology, current limits and possible new horizons for probabilistic modeling of historical settlement systems. To do this, our paper examines two studies, foreign and domestic, devoted to modeling the systems of historical settlement of different eras. In particular, it is pointed to the critical dependence of the approach, presented in these studies, on the scale and uniformity of the conditions of the territory under consideration. The way out of this situation is presented to the author of this paper by involving new types of data from historical sources for modeling, revising and rejecting unnecessary assumptions about the specific nature of the settlement process.

Keywords: probabilistic modelling, bronze age, quadrat analysis, random variable, Poisson distribution, chi-squared test, rural settlement system, pyatina, pogost, cadastral book

Abstract: The article compares the author's work on modeling historical processes, completed 30 years ago, with the publication of a similar model devoted to some hypotheses of the penetration of the “steppe heritage” into the gene pool of European farmers of the Bronze Age. The changes that have occurred in the computer modeling of historical processes, which significantly increase their research potential, are characterized. On the example of the models under consideration, it is discussed in what aspects they help to clarify and develop knowledge about the past. The features of spatially explicit models, which are a subset of imitation models (simulations), are demonstrated, including the possibility of taking into account the genetic parameters of the modeled populations and subsequent comparison with aDNA. The degree of novelty of the results obtained in the two compared models is estimated. It is noted which results of the latest research and additional considerations should be taken into account in order to further bring the behavior of the model closer to reality. It is suggested that the modeling results make it possible to associate the origin of the “steppe heritage” not so much with the population of the Yamnaya culture, but with the contact zone in the south of Ukraine, where interaction with the late Trypillian and other agricultural cultures took place. The necessity of supporting with models any assumptions about historical processes that does not follow directly from empirical material in order to confirm the fundamental reproducibility of the proposed scenario is substantiated.

Keywords: simulation, spatially explicit model, bronze age, Europe, corded ware, Yamnaya culture, Trypillian culture, aDNA, migration, plague

Abstract: The author develops a mathematical-statistical approach to the problem of estimating the size of Genoese medieval population in Byzantium. The data source is notarial acts covering commercial partnerships, freightage, wills, purchase and sale of houses, goods and people drawn up in the Genoese colony of Constantinople at the end of the 13th century. The will form has a fairly uniform structure. In addition to the mandatory record of names of the contracting parties and witnesses of the transaction, it may also register names of the third parties. Thus, these data on the clientele of Genoese notaries represent a dataset which may indirectly indicate the size of the entire trading Genoese community of Byzantium. This approach is based on a constructed formalized model that describes the behavior of merchants when visiting and concluding a transaction attested by a notary. This makes it possible to pass in a natural way from the initial to the statistical problem of estimating the size of a finite aggregate and use this mathematical theory for its calculation. In this case, the author applies the approach associated with the use of the maximum likelihood function that is a novelty. The resulting formula allows (with a certain degree of probability) one to estimate the required size of the Genoese population. It is interesting that this estimate, on the whole, coincides with the result of A.L. Ponomarev obtained earlier for the same problem using Zipf's empirical law.

Keywords: size of a finite population, sample, statistical estimation, ranged series, notarial acts, Genoese, Constantinople, multinomial distribution, maximum likelihood estimation, Zipf’s law

Abstract: This paper outlines factors that impacted the change of demographic behavior of the rural population of the Central Black Earth Region from the middle of the 19th century to the late 20th century. Factor scores have been determined when modeling demographic processes in 1,500 rural settlements. The authors rely on a classical view that late traditional society was highly dependent on negative natural events, characterized by low migration and strong ties of birth rates with death rates. It has been shown that extraordinary positive factors (the rise of zemstvo medicine in particular) stimulated unlimited natural growth. Short-term stress factors (wars, hunger and epidemics) led to compensatory reproduction. The authors also come to a conclusion that modernized society (the latter half of the 20th century) had more complicated relations between demographic behavior factors. The evolution of modernized micro-communities (individual settlements) was subjected to a great number of nonlinear effects. The data demonstrate that the agrarian society at the time continued the move to natural growth although to a lesser extent than during the previous stages. However, the results of such an intention were largely neutralized by migration.

Keywords: migration, mortality, birth rate, Central Black Earth region, agrarian population, historical demography, computational history, social simulation, demographic transition, demographic strategies

Abstract: The article proposes an improved model of St. Petersburg Stock Exchange index dynamics and constructs a similar model of Helsinki Stock Exchange index on the basis of published results of a counterfactual model predicting the hypothetical dynamics of St. Petersburg Stock Exchange index after July 1914 under the assumption that there is no war. The author hypothesizes that internal economic factors that determined the downward trend of St. Petersburg Stock Exchange index also influenced the dynamics of Helsinki Stock Exchange index under the assumption that there was no war. To test this hypothesis the author has constructed (in the R software environment) the ARIMA statistical model that is an integrated autoregressive-moving average model which extends the ARMA model for non-stationary time series. The constructed counterfactual models proved that while the influence of pre-war factors remained, the dynamics of both indices did not show similar trends thus suggesting that the Finnish stock market was developing without any noticeable look at St. Petersburg Stock Exchange and inner economic factors of the Russian Empire.

Keywords: stock exchange index, St. Petersburg Stock Exchange, Helsinki Stock Exchange, the Grand Duchy of Finland, Russian Empire, economic history, World War I, counterfactual model, ARIMA model, software environment R

Abstract: The article presents the results of modeling demographic strategies of 1.5 thousand Tambovskaya Guberniya rural settlements in 1959-1989. The authors consider methodological and tool aspects related to the formation of a fractal model and computer experiments based on it. They have determined the values of governing factors affecting the formation of collective goal-oriented ideas about the expected migration rate and the rate of population's natural increase or decline. The results obtained can arguably be extended to a few similar regions of Russia. This study is sequel to previous works where models for the period from the middle of the 19th century to the middle of the 20th century have been set forth. Modeling has demonstrated that there are two groups of settlements: migration donors and migration recipients. During the period understudy the rural population (when setting their life goals) relied mostly on migration which evidently was a depressing agent for natural increase. Nevertheless, the majority of peasant societies were open to a certain natural increase. .

Keywords: fertility and mortality, migration, late soviet period, fractal geometry, computer model, Russian peasantry, demographic strategies, historical demography, social simulation, Central Black Earth

Abstract: The article considers state elite of the Russian Empire in the 19th century consisting of civil servants of the 4th-1st classes. Special attention is paid to its interaction with the system of civil ranks promotion. The author studies the way the Table of Ranks and terms in office system favored inter-elite rivalry for the influence on the emperor. The latter is the main element of the civil ranks promotion system whose formal sanction influenced career dynamics of civil servants in the upper strata. The interactions between the monarch, top civil servants and the formal ranks promotion system formed the dynamics of annual promotion of some contenders to the 4th civil rank. This dynamic row is considered through the lens of self-organized criticality theory. The author hypothesizes that the actions of the state authorities aimed at filling the deficit of mid-ranking civil servants resulted in abundant numbers of civil servants of the 5th and upper classes and low control over the career dynamics of the upper civil servants and the system of top ranks promotion was in the state of criticality. Analysis of the dynamics of annual ranks promotion to the 4th class supports this idea.

Keywords: theory of self-organized criticality, political modernization, civil service, overproduction of the elite, Table of ranks, nineteenth century, the Russian Empire, officialdom, bureaucracy, mathematical methods

Abstract: Mathematical modeling is increasingly popular to study spatial distribution of population. This trend is determined by the necessity to study the dynamics of population quantity and density, reconstruct lacking data, search for population displacement factors and evaluate their influence on this process. The current study develops the model describing the system of centers distribution (the settlement structure) of the one-level hierarchy. Such a modeling aims at searching for basic factors and evaluating their influence on the process, associating optimum parameters which characterize the system of centers distribution. The model developed by the author is based on the innovative approach of S.M. Gusein-Zade which makes it possible to describe the regularities in the location (distribution) of heterogeneous settlements and other centers on the territories. The variations calculus methods applied as well as the concept of Pareto optimality allows one to find parameters of the existing systems of centers location and quantitatively describe the phenomena occurring in them. The model has been tested to study a medieval rural settlement and has proven adequate and correct. It has been demonstrated that the process of free population displacement was caused by two counterbalancing factors: a centripetal one (concentration of settlements around their centers) and a centrifugal one (displacement of population near the farmland).

Keywords: Pareto equilibrium, calculus of variations, human settlement system, scribe book, central location theory, Poisson distribution, Mathematical modelling, population density function, functional, linear regression

Abstract: The article describes how the author forms a counterfactual model forecasting the dynamics of Saint Petersburg Stock Exchange index after July 1914 before the war which radically changed the factor role of prewar dynamics. The author hypothesizes that the decreasing trend of St. Petersburg Stock Exchange index during the last prewar year was caused by internal economic factors which could determine further dynamics of the index when no war is assumed. To test this hypothesis the author has developed ARIMA statistical model within the R software environment. This model is an integrated model of autoregressive moving average which is an extension of the ARMA model for nonstationary time series. The counterfactual model has demonstrated that in case the influence of the pre-war period factors continued, the dynamics of the index over the next year would tend to decrease, even if the war did not begin. Thus, one can speak of the beginning recession phase in the cyclical development of Russian industry in 1913.

Keywords: the counterfactual model, mathematical modeling, the stock exchange index, the St. Petersburg Stock Exchange, the First World War, Russian empire, economical history, retro forecast, the ARIMA model, the R software environment

Abstract: This work crowns a series of studies addressing the forms of ceramics within the Eastern Europe’s steppe belt in the Bronze Age. These studies aimed at searching for the internal structural arrangement of complexes of ceramics forms as a complicated system with its own laws of development. The form of a vessel was represented as a set of quantitative markers describing its profile. The study covers vessel complexes belonging to Srubnaya, Sintashtinskaya, Dono-Volzhskaya Abashevskaya cultures, cultural formations of KMK circle as well as Pokrovskiy cultural type. When studying complexes of ceramics forms the authors employed cluster analysis with subsequent consideration of the group size correlation diagram and non-compactness ratio of the groups. Structural arrangement of complexes of ceramics forms has been found for a number of cases. The first arrangement corresponds to a reverse power (“hyperbolic”) law dependence on the correlation diagram whereas the second one corresponds to a logarithmic dependence. For complexes with such structural organization the term "archaeological cultural type" (ACT) is proposed. It is shown that ACT has its own "life cycle" consisting of three periods. These are the advent, the hey-day and the decay. 

Keywords: Pokrovsk cultural type, Dono-Volzhskaya Abashevo culture, bronze age, life cycle, archaeological cultural type, cluster analysis, forms of ceramics, Sintashta culture, Timber-grave culture, post-catacomb cultural entities

Abstract: The article gives an example of the way self-organized criticality (SOC) theory can be used to study a number of nonlinear effects in social media, protest movement sharp rises in particular. The authors describe the key ideas of the SOC theory and methods to identify pink noise as a SOC attribute. The research object is an associated total of Facebook communities which lent informational and organizational support to the impeachment of Brazil’s president Dilma Rousseff. Chronological frames of the work is January 2015 – august 2016. It has been shown that net activity of some communities understudy was a pink noise within a number of periods. SOC systems can demonstrate explosive activity increase that is the inner potential of systems for self-destruction and/or qualitative transformation. The protest social network studied had key features to generate informational avalanches – fast and big explosions of information creation, transmission and copying. The authors name facts supporting the idea that SOC advent in the nets is caused by mass street actions (violent as well). One parameter of the numerical series studied is a power law factor which can be indicative for network diagnostics, in particular, for revealing their potential for spontaneous informational avalanches.

Keywords: non-linear effects, self-organized criticality, pink noise, avalanches, protests, Facebook communities, Brazil, Dilma Rousseff, impeachment, social networks

Abstract: The article presents the results of analysis of long time data series reflecting the intensity of peasant disturbances in various guberniyas of European Russia in the second half of the 19th century by means of self-organized criticality theory . The types of peasant protests are presented that depend on the regularity of protest intensity fluctuations in different guberniyas. It has been found that the population density and the average allotment of land exerted the most powerful influence on the formation of protest potential. A hypothesis has been proposed and substantiated that the presence of pink noise in changes of peasant disturbances intensity means that most of regional communities were in a critical state. This demonstrates serious risks of system stability throughout the period understudy. Inner social environment and structural features existed that led to leaping fluctuations and unexpected increase of peasant disturbances. Regional communities were prone to uprisings under the influence of quite usual factors.

Keywords: Russia , the theory of self-organized criticality, spectral analysis, pink noise, correlation analysis, social processes, history, cluster analysis, interdisciplinary studies, peasant disturbances

Abstract: Dynamics of the strike movement in Russia in the late 19th – early 20th centuries is analyzed. Multiple statistical data on workers’ fighting for their rights in this period provides for effective use of mathematical modeling to better understand the causes for workers’ protest activity increase that was especially high in 1903. In particular, one can evaluate the role of activists in the strike movement development as well as the influence of repressions on activists’ popularity. The mathematical model by Andreev A.Y. and Borodkin L.I. has been modified to describe the interaction of authorities, activists and workers. The use of the modified model to describe the strike movement dynamics is given as an example. To test the model’s adequacy the calculation data were compared with corresponding statistical indicators. The results demonstrate that the model is rather good at describing the changing strike activity in Russia in the late 19th – early 20th centuries. When the influence of activists is generally low (1895-1899), it accounts for 10-20% of the strike movement. When it reaches the critical point, the role of activists in the strike movement growth increases sharply and becomes dominating in 1902 when this point is overcome. The model is also correct at describing the general trend of decreasing repressions by the authorities and the growth of activists’ influence.

Keywords: statistical indicator, workers' strain, stressors, interaction of social groups, activists' influence, authorities' strain, critical value, normalized indicator, mathematical model, strike movement