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## The mathematical model for predicting the earliest diagnostics period of the secondary distant metastases growth process of breast cancer

Previously, the mathematical models (CoMPaS and CoM-III) of primary tumor (PT) growth and secondary distant metastases (sdMTS) growth of breast cancer (BC) considering TNM classification have been presented (Tyuryumina E., Neznanov A.; 2017, 2018). Goal: To detect the earliest diagnostics period of visible sdMTS via CoMPaS and CoM-III.

The design of logistics chain is strategically important issue for almost every company. There is a large variety of model formulations and solving algorithms, which vary in fundamental assumption, mathematical complexity, etc. In this paper classification of models of dislocation of logistic capacities is considered and the multicriteria model of optimum dislocation of warehouses in supply chains is offered.

This study is an attempt to obtain reliable data on the natural history of breast cancer growth. The opportunities for using classical mathematical models (exponential and logistic tumor growth models, Gompertz and von Bertalanffy tumor growth models) were analysed in order to describe growth of the primary tumor and the secondary distant metastases of human breast cancer. Our results suggest a new «Consolidated mathematical growth Model of the Primary tumor and the Secondary distant metastases» (CoMPaS). The CoMPaS is based on exponential tumor growth model and consists of a system of determinate nonlinear and linear equations. The CoMPaS describes correctly the primary tumor growth (parameter T) and the secondary distant metastases growth (parameter M). Also, CoMPaS associates with data of 10–15-year survival in patients with the different tumor stage. Analysis of the metastases «nonvisible period» growth indicate the case of discrepancy between 15-year survival depending on tumor stage. In conclusion, the CoMPaS and supporting computer program were build to improve the accuracy of the forecast on survival of breast cancer and facilitate the optimisation of diagnosing secondary distant metastases. This led to completely original results that show how the growth rate of the metastases can change in relation to the growth rate of the primary tumour, taking into consideration its size and diameter of the tumour.

The search for novel parameters to predict the risk of relapse in breast cancer was conducted. Significant correlation between the risk of relapse and α-2A adrenergic receptor (*ADRA2A*) expression was revealed using public microarray datasets. This relationship was confirmed by validation on independent microarray dataset. It was found that when assessing the risk of BC relapse, the accuracy of prediction based solely on the expression of *ADRA2A* gene is close to that made using OncotypeDX and MammaPrint test systems. In this case, addition of only one or two supplemental prognostic markers (for instance, expression of *SQLE* gene or *SQLE* and*DSCC1genes*) to *ADRA2A* ensures the accuracy of prediction not inferior to reliability of these test systems.

Proceedings of the III International Conference in memory of V.I. Zubov "Stability and Control Processes (SCP 2015)".

Nowadays the random search became a widespread and effective tool for solving different complex optimization and adaptation problems. In this work, the problem of an average duration of a random search for one object by another is regarded, depending on various factors on a square field. The problem solution was carried out by holding total experiment with 4 factors and orthogonal plan with 54 lines. Within each line, the initial conditions and the cellular automaton transition rules were simulated and the duration of the search for one object by another was measured. As a result, the regression model of average duration of a random search for an object depending on the four factors considered, specifying the initial positions of two objects, the conditions of their movement and detection is constructed. The most significant factors among the factors considered in the work that determine the average search time are determined. An interpretation is carried out in the problem of random search for an object from the constructed model.The important result of the work is that the qualitative and quantitative influence of initial positions of objects, the size of the lattice and the transition rules on the average duration of search is revealed by means of model obtained. It is shown that the initial neighborhood of objects on the lattice does not guarantee a quick search, if each of them moves. In addition, it is quantitatively estimated how many times the average time of searching for an object can increase or decrease with increasing the speed of the searching object by 1 unit, and also with increasing the field size by 1 unit, with different initial positions of the two objects. The exponential nature of the growth in the number of steps for searching for an object with an increase in the lattice size for other fixed factors is revealed. The conditions for the greatest increase in the average search duration are found: the maximum distance of objects in combination with the immobility of one of them when the field size is changed by 1 unit. (that is, for example, with 4x4 at 5x5) can increase the average search duration in e^1,69≈5,42. The task presented in the work may be relevant from the point of view of application both in the landmark for ensuring the security of the state, and, for example, in the theory of mass service.

We propose a new mathematical growth model of primary tumor and primary metastases which may help to improve predicting accuracy of breast cancer process using an original mathematical model referred to CoM-IV and corresponding software. The CoM-IV model and predictive software: a) detect different growth periods of primary tumor and primary metastases; b) make forecast of patient survival; c) have higher average prediction accuracy than the other tools; d) can improve forecasts on survival of BC and facilitate optimisation of diagnostic tests. The CoM-IV enables us, for the first time, to predict the whole natural history of primary tumor and primary metastases growth on each stage (pT1, pT2, pT3, pT4) considering only on primary tumor sizes. Summarising: CoM-IV a) describes correctly primary tumor and primary distant metastases growth of IV (T1-4N0-3M1) stage with (N1-3) or without regional metastases in lymph nodes (N0); b) facilitates the understanding of the appearance period and manifestation of primary metastases.

This paper is devoted to mathematical modelling of the progression and stages of breast cancer. The Consolidated mathematical growth Model of primary tumor (PT) and secondary distant metastases (MTS) in patients with lymph nodes MTS (Stage III) (CoM-III) is proposed as a new research tool. The CoM-III rests on an exponential tumor growth model and consists of a system of determinate nonlinear and linear equations. The CoM-III describes correctly primary tumor growth (parameter T) and distant metastases growth (parameter M, parameter N). The CoM-III model and predictive software: a) detect di erent growth periods of primary tumor and distant metastases in patients with lymph nodes MTS; b) make forecast of the period of the distant metastases appearance in patients with lymph nodes MTS; c) have higher average prediction accuracy than the other tools; d) can improve forecasts on survival of breast cancer and facilitate optimisation of diagnostic tests. The CoM-III enables us, for the rst time, to predict the whole natural history of PT and secondary distant MTS growth of patients with/without lymph nodes MTS on each stage relying only on PT sizes.

This book deals with mathematical problems arising in the context of meteorological modelling. It gathers and presents some of the most interesting and important issues from the interaction of mathematics and meteorology. It is unique in that it features contributions on topics like data assimilation, ensemble prediction, numerical methods, and transport modelling, from both mathematical and meteorological perspectives.

The derivation and solution of all kinds of numerical prediction models require the application of results from various mathematical fields. The present volume is divided into three parts, moving from mathematical and numerical problems through air quality modelling, to advanced applications in data assimilation and probabilistic forecasting.

The book arose from the workshop “Mathematical Problems in Meteorological Modelling” held in Budapest in May 2014 and organized by the ECMI Special Interest Group on Numerical Weather Prediction. Its main objective is to highlight the beauty of the development fields discussed, to demonstrate their mathematical complexity and, more importantly, to encourage mathematicians to contribute to the further success of such practical applications as weather forecasting and climate change projections. Written by leading experts in the field, the book provides an attractive and diverse introduction to areas in which mathematicians and modellers from the meteorological community can cooperate and help each other solve the problems that operational weather centres face, now and in the near future.

Readers engaged in meteorological research will become more familiar with the corresponding mathematical background, while mathematicians working in numerical analysis, partial differential equations, or stochastic analysis will be introduced to further application fields of their research area, and will find stimulation and motivation for their future research work.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.