APPLIED MATHEMATICS
AND CONTROL SCIENCES

The development of computer models of orthopedic products allows adding to the arsenal of a trauma surgeon a digital toolkit that allows to calculate the biomechanical consequences of the chosen tactics of reconstructive and restorative treatment. Thus, when using percutaneous osteosynthesis systems with the help of a digital twin of the external fixation apparatus, the operating specialist at the preoperative stage can assess the influence of the choice of the apparatus layout on the risk of possible separation of bone fragments.Clinical indications for the use of the Ilizarov apparatus of the Bagirov modification are fractures of diaphyses of the tibia bones, which account for up to 45 % of all fractures of long bones of the human skeleton. Fracture healing is accompanied by a large number of complications, so the problem of improving the results of treatment of patients with this injury is still relevant for modern traumatology and orthopedics. The use of external fixation devices makes it possible to ensure stable functional osteosynthesis and eliminate the need for plaster immobilization of the segment or limb in the postoperative period. The use of compression-distraction devices allows us to expand the potential use of external fixation devices to solve the problems of distraction osteogenesis. Biomechanical relationships in the bone-apparatus system are a significant factor that allows objectifying the apparatus layout and the mode of motor rehabilitation in the early postoperative period.Within the framework of this study, a mathematical model of the Ilizarov apparatus of the Bagirov modification was constructed and the stress-strain state of the apparatus components under model loads was calculated using the finite element method. To validate the computer model, a field experiment was conducted on a Walter+Bai AG LFM-5 universal testing machine. The investigated structure was subjected to axial compression with a load up to 1000 N. As a result of the comparative analysis of the resistance of the device to axial compression, it was found that the results of finite element calculations describe the experimental results with sufficient accuracy. Comparison of the calculation results with experimental data allows us to state that the proposed computer model correctly describes the mechanical behavior of the studied medical device and can be used in computational experiments to assess the functionality of various device layouts.

The least absolute deviation method is one of the most common alternatives to the least squares method in regression analysis. It allows obtaining stable estimates of coefficients when the probability density of random errors has more elongated tails compared to the normal distribution. However, when several violations of the Gauss-Markov conditions are combined, for example, with a one-sided nature of outliers and the presence of a correlation between explanatory variables and random errors, the least absolute deviation method also does not provide acceptable accuracy in estimating regression dependencies. One of the promising ways to solve this problem may be the weighted least absolute deviation method.The problem of determining the parameters of linear regression models based on the weighted least absolute deviation method is considered. Exact algorithms for solving it are described. The computational efficiency of exact algorithms for solving the problem of the weighted least absolute deviation method is investigated. It is proved that adding weight coefficients to the coordinate-wise and modified gradient descent algorithms did not cause changes in terms of computational complexity and accuracy of the solution. Nevertheless, a small increase in the execution time of computational experiments was recorded due to the addition of an additional operation to the descent algorithms. This dependence is more noticeable in the coordinate variant. This is due to the fact that the value of the objective function in it is determined at each nodal point of the nodal straight line up to finding the minimum, while in the gradient descent it is determined only at the point of extremum.As a result of the comparative analysis with the gradient projection method and solutions of direct and dual problems of linear programming using the simplex method, it was found that they are more than an order of magnitude inferior to the gradient descent along nodal straight lines in terms of computation time. It is shown that the gradient projection method does not guarantee finding an exact solution to problem.

The authors propose a model of the testing process in the notation of colored Petri nets. Before starting the simulation, the information flows of the process of mastering the discipline «Informatics» in IDEF3 notation by students of the Department of Applied Information Technologies of the ION RANEPA were studied. In the process of building the model, the following task was solved:let's give a Petri net consisting of a set of positions P = {p1, p2, ..., pn} and a set of transitionsT = {t1, t2, ..., tm}. The network simulates the testing process. Then there are such states of positions P that the following conditions are met for each transition T in the network:1. ∀ti ∈ T: (∀pi' ∈ Pred(tj): pi'(pj) = 1) → T(P) = 1, where Pred(tj) is the set of all preceding transition positions ti. 2. ∀tj ∈ T: tj(pj) = 1 → (∀p' ∈ Succ(tj): pj'(pj) = 1), where Succ(tj) is the set of all the following transition positions tj.That is, it is necessary to formalize the educational process using a colored Petri net. Find a labeling that can be achieved in a given colored Petri net that satisfies the following condition: there is a sequence of transitions (transactions) starting from the initial labeling, such that each subsequent labeling is obtained by applying the corresponding transition to the previous labeling. I.e.: ∃ M0: (initial labeling) → M*, where → M* denotes the reflexive and transitive closure of the transition relation in the Petri net.Solving the problem of the reachability of network labeling gave the concept of an optimal scenario for monitoring the development of competencies in the discipline: Some tasks and questions do not have to be included in the testing. Various scenarios for monitoring student progress are presented using the example of discipline testing. For example, one of the variants of the test scenario assumed the presence of a common resource with questions and assignments on the five topics under study. There were 50 questions in the database. The answer to each question can be rated from zero to 10 points. The test is considered passed if the student scored at least 41 points. The numbers of completed tasks and questions with the correct answer were also recorded.Another test option: 10 questions for each topic and the response time to the question (task) was recorded.A matrix analysis of the model was carried out. The vector of the initial marking μ(pi) = {47,1,1,1,0,0,0,0} and μk = {30, 0, 0, 0, 0, 0, 0, 41} at the end of testing. The analysis of the reachability of the μk marking allowed us to obtain one of the sequences of triggering transitions U k-1 = (t1, t3, t4, t3, t3, t4). Where transitions t1 – t4 fix the beginning and end of an elementary process (for example, the beginning of testing – the end of testing).The analysis of the model allowed us to investigate the dynamics of studying the topics of the discipline « Informatics »: how many times the student returned to re-reading the module materials, how many times he turned to additional material. Monitors attached to the transitions made it possible to determine the time for each operation and draw up protocols for studying the discipline.

The application of fuzzy modeling methods to analyze the effectiveness of investment instruments is considered. When choosing a financial strategy in a situation of uncertainty, such an analysis helps to evaluate and make a decision. Since the parameters of the financial system in conditions of uncertainty can be impossible to establish accurately, problems arise that are described by a number of characteristics of an indistinct nature. In the work, a system of indicators was set for evaluating the investment strategy. The significance of each indicator is determined using weighting coefficients, which are determined using the method of paired comparisons and the Saati scale. To describe financial instruments, special linguistic variables were introduced, for each of which term sets were set. Each term is a fuzzy trapezoidal number.After fixing the current values characterizing the financial system, a fuzzification procedure is performed, that is, the introduction of fuzziness. Then, in a certain way, the convolution operation is performed for all levels of the model indicators, taking into account the weighting coefficients of significance. As a result, we obtain a general aggregated characteristic of an investment instrument, according to which it is possible to draw a conclusion about the level of its effectiveness. The application of the obtained results is demonstrated by examples.