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So far, there are no efficient algorithms to solve a problem of finding the low degree annihilators for arbitrary Boolean function. In the paper we present a new algorithm to find affine annihilators for an arbitrary Boolean function. We start with considering the identity fg ≡ 0 or the arbitrary Boolean function f and its possible affine annihilator g. We use a special representation of the Boolean function in sum of its sub-functions to reduce degrees of considering functions in previous identity. As a result, we establish equivalence between the identity fg ≡ 0 for Boolean functions of n variables and the system of Boolean equations of n-1 variable.An algorithm for finding the affine annihilators for the arbitrary Boolean function f must find all the affine functions g so that fg ≡ 0. Our algorithm is based on reducing the problem of finding the affine annihilators for the Boolean function f of n to the similar problem for its sub-functions of n-1 variable. The presented algorithm has the following advantages:An input function can be presented in different ways;Output can be also presented in different ways;The algorithm can be effectively parallelized.It should be noted that the result we have obtained is not final and highlights some development directions: first, to study the impact of its input and output on the efficiency of the algorithm of various representations and, second, to use our idea of constructing the algorithm for development of algorithms, which allow finding the 2nd, 3rd, etc. degree annihilators for a specified Boolean function.

A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.

The article investigates the problems of reduction (decomposition) of multidimensional data models in terms of hypercube OLAP-structures. Describes the case when a data structure is defined by the array that slices and dices the hypercube into the odd number of subcubes, and this set of subcube structures becomes decomposed. Defines an exact upper bound for increasing a computational performance of methods to analyze OLAP-data on subcubes, which determines the decomposition approach efficiency in comparison with the OLAP-data analysis on a complete unreduced hypercube. A compared efficiency of the hypercube decomposition into two subcubes on the sets consisting of the even and odd number of subcube structures has shown that with considerable data partitioning for methods of a polynomial complexity degree the decomposition efficiency essentially is independent on this factor and rises with increasing complexity degree of methods applied.When using the mathematical methods to study decomposition (reduction) of large hyper-cubes of multidimensional data of analytical OLAP systems into subcube components, there is a need to find conditions for minimising the computational complexity of methods to solve the problems of the OLAP hyper-cube analysis during data decomposition in comparison with using these methods for analyzing large amounts of information that is accumulated directly in the hyper-cubes of multidimensional OLAP-data to establish the criteria for decreasing or increasing computational performance when applying methods on the subcube components (reduction methods) as compared to applying these methods on a hypercube (non-reduction or traditional methods), depending on one or another degree of complexity of complex methods.The article provides an accurate quantitative estimate of decreasing computational complexity of reduction methods for analyzing OLAP-cubes as compared to the non-reduction methods in the case when said methods have the polynomial complexity and the original hypercube array of data comprises the odd number of subcubes.

This article presents a clinical case of successful treatment of gram-negative sepsis in a patient with HIV-associated non-Hodgkin’s lymphoma. The patient was admitted to the intensive care unit in critical condition after the third course of polychemotherapy according to the ICE scheme. The severity of the condition was due to nosocomial pneumonia, septic shock, multiple organ failure, immunosuppression against the background of PCT and HIV infection, and the lack of specific treatment for HIV infection. Despite the absence of a positive blood culture throughout the entire treatment period, the diagnosis of sepsis was not in doubt, according to the criteria of the 2001 International Consensus Conference on Sepsis. The cause of the septic state was the combined effect of bacteria (Pseudomonas aeruginosa) and fungi (Candida albicans, Candida krusei) against the background of persistent HIV infection. The patient’s pneumonia was destructive and was twice aggravated by spontaneous pneumothorax. At the initial stage, intensive therapy led to positive dynamics. The severity of the systemic inflammatory response decreased, the acute respiratory insufficiency regressed, the X-ray pattern improved, and laboratory parameters stabilized. Despite the continued intensive therapy in the former volume on day 19, a sharply negative dynamics was noted, which led to a re-transfer of the patient to artificial ventilation of the lungs. The replacement of antibiotics and the specific treatment of HIV infection led to the patient’s recovery. In the future, chemotherapy was not carried out. Remission of the disease lasts six years.

Classical multiphase filtration models, which are based on Darcy's law, are well studied and actively used in modern oil engineering. However, such models do not allow efficient describing of processes with power-law memory effects or with spatial non-locality effects. In recent years, there has been a significantly increasing interest in models with fractional derivatives and integrals that allow such effects to be taken into account.The article considers the space-fractional generalization of the two-phase Black Oil model. This model is built on the basis of a fractional modification of the Darcy’s law with the Riesz potential, which is one of the possible generalizations of a fractional integral to the case of a multidimensional space. The use of such a modification of Darcy's law allows efficient describing of filtration processes in heterogeneous fractured porous media with the effects of spatial non-locality.For the numerical solution of the system of equations from the obtained model, the use of the IMPES method is proposed. For this purpose, from the presented system of equations, one equation is selected that describes the evolution of pressure. This equation is written provided that the capillary pressure variation within the time step is neglected.A special case of the pressure equation with the Riesz potential of radial function is considered, which describes the pressure variation in the case of a radial plane flow. For this equation, a self-similar solution is constructed using the Mellin integral transform method. A representation of this solution in the form of the Mellin-Barnes contour integral was obtained, which made it possible to write it in terms of the Fox functions. It is shown that in the limiting case of a zero degree of the Riesz potential, this solution coincides with the self-similar solution of the classical heat equation. The constructed self-similar solution can be further used in the software implementation of the numerical solution of the model presentedThe main line of further research is to develop and implement a software computing system based on the proposed fractional generalization of the two-phase Black Oil model.

The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.

Тhe main treatment for angle-closure glaucoma is laser iridectomy. Somatic comorbidity (autoimmune, allergic, chronic infectious diseases) affects the results of laser iridectomy.Purpose: to evaluate in practice the algorithm for applying the method of laser iridectomy in compliance with preventive measures, including medical support before and after surgery, both in patients with concomitant somatic pathology and without it.Patients and methods. The analysis of combined laser iridectomy’s results in 61 patients (94 eyes) was carried out. A method was used to determine the tactics of management the patients with latent stage of angle-closure glaucoma and pigment dispersion syndrome (Patent RU N 2726404), including an assessment of the general immune status based on information about somatic diseases and medications taken. Combined laser iridectomy was performed in two stages in one visit. The first stage is coagulation in the projection of the lacunae at the periphery of the iris; at the second stage, two through holes were formed. All patients received medication in accordance with the algorithm. The level of reactive hypertension was assessed in 1 hour after surgery, on the first and seventh days after the intervention.Results. In case of standard medical support for laser iridectomy, the level of reactive hypertension is higher in the group of patients with concomitant somatic pathology than without it. Adding additions to the standard drug regimen in a group of patients with impaired general immune status ensures that the target intraocular pressure is achieved. The use of an algorithm for performing laser iridectomy in patients with angle-closure glaucoma and concomitant somatic pathology makes it possible to achieve the target intraocular pressure early after surgery and to avoid complications.

Social networks have firmly entered the lives of billions of global Internet users worldwide. They communicate in social networks, play online games, make purchases, organise online events — exchange content from all walks of life [1, 2]. The most popular and well-known services in Russia are Vkontakte (vk.com), Youtube.com, Facebook.com, Odnoklassniki (Ok.ru), etc. The interfaces of such platforms allo — fake accounts. In this paper, we propose an approach to detect bots using the LiveJournal social network as an example. For this, we investigated the characteristics of the user’s egograph and performed a comparative analysis of the results of the classification algorithms.

A novel mathematical apparatus allowing formulation and justification of a new approach towards the setting of the mathematical problem of band-pass layered dielectric filters (LDF) synthesis is developed. Аrbitrary layered dielectric systems with piecewise continuous physical media parameters given by the functions of dielectric permittivity and of magnetic permeability, both depending on the coordinate along the normal to the layer pile, with fixed discontinuity points of at least one of the mentioned functions are examined. For such systems, an important conservation law for the difference of the squares of absolute amplitude values of plane waves propagating left and right in given layered medium is stated, which further leads to the traditional energy conservation law in lossless layered media. This new identity law allows turning from synthesis problems in terms of fractional rational energy reflectivity and transmittance of layered systems to equivalent tasks for profiling functions introduced in the work, representing only the numerator or only the denominator of the expressions usually considered in the synthesis. A new concept of the feasible ideal is introduced for the energy coefficients of reflection and transmission of layered systems. It is shown that the feasibility of the energy coefficients of reflection and transmission of layered systems is equivalent to the feasibility of the profiling functions of such systems, which together with the main identity allows a significant change of the existing LDF synthesis approach. The rule for converting the ideal of the reflection or transmission coefficient into the ideal of the profiling function is given. The proposed synthesis problem statement leads to considerably less intensive computational procedures.

Macular holes are the most often pathology leading to reduced vision among persons after 60 years. The macular hole is the acquired disease which is a defect of a foveola part of retina on all thickness from an internal limiting membrane to an external segment of a photoreceptor layer. The standard method of treatment is surgery method: a microinvasive vitrectomy with membrane peeling in macular area. There is a number of the techniques increasing efficiency of operative intervention. Even in case of successful surgical intervention macular holes are not always closed.The purpose: development and assessment of morphological parameters of the central retina with a new surgical method of closing earlier operated macula hole using a free flap.Material and methods. 40 patients (40 eyes, 28 women and 12 men) had participated in this study. All patients were operated about macular hole with negative anatomic result. By technology which were used earlier, patients were divided into 3 studied groups (with PRP mass using, with introverted flap using, with mechanical matching of edges of a hole). The main method of research was the optical coherent tomography. All patients were operated by an original technique. It’s purpose was a closing of earlier operated rigid macular hole using a free flap.Results. The defect of a photoreceptor component was closed with an additional expansion of earlier created macular rexis in patients of the 1st and 3rd groups at reoperation. The earlier created flap of an internal limiting membrane was used in patients of the 2nd group. In the analysis of dependence of change of visual acuity from morphological parameters of a macular hole after primary surgical treatment the inverse correlation relation from diameter of a macular hole on small and big radiuses is revealed. The highest visual acuity after a reoperation was reached at patients of the second group.Conclusions. It is necessary to consider the technique which was used earlier, diameter of a macular hole on small and big radiuses, postoperative edema in a parafovea zone for planning the surgical treatment and a forecast of functional results.

The method of phacovitrectomy can be used for surgical treatment of patients with cataract and asteroid hyalosis. However, there is still a topical question in the choice of surgical treatment taking into account the completeness of the posterior hyaloid membrane (PHM) detachment.Patients and methods: the study involved 24 patients (24 eyes) with cataract and asteroid hyalosis. The average age of patients was 72.9 years, out of the total number of men — 11, women — 13. Pre-and postoperative diagnosis of all patients included standard methods of investigation, as well as optical coherence tomography performed on the Cirrus HD-OCT 400, Carl Zeiss, Germany. Pre-and postoperative subjective satisfaction was assessed using a validated visual Function Qusetionnaire 14 ophthalmological questionnaire. Patients were divided into 3 groups. In the first group of patients, cataract phacoemulsification with posterior chamber lens implantation and total three-port vitrectomy through a flat part of the ciliary body were performed. In the second patients’ group, cataract phacoemulsification with posterior chamber lens implantation and subtotal vitrectomy through posterior capsulorexis were performed. In the third group of patients, cataract phacoemulsification with posterior chamber lens implantation was performed. All operations were performed with the help of ophthalmic systems Infinity Vision System (Alcon, USA) and Constellation Vision System (Alcon, USA).Results: in the first day after the operation, all patients noted improvement in visual acuity and quality of life, but more significant improvement was noted in patients of the first and second groups.Conclusion: analysis of the results of surgical treatment, questionnaire survey, preoperative diagnosis allowed to develop an algorithm for choosing the method of surgical treatment in patients with cataract and asteroid hyalosis, taking into account the state of PHM.

A general model of the origin and evolution of chaotic wave processes in electrochemically active neuronal media based on the proposed method of matrix decomposition of operators of nonlinear systems has been developed. The mathematical models of Hodgkin – Huxley and FitzHugh – Nagumo of an electrochemically active neuronal media are considered. The necessary conditions for self-organization of chaotic self-oscillations in the FitzHugh – Nagumo model are determined. Computer modeling based on the matrix decomposition of chaotic wave processes in electrochemically active neuronal media has shown the interaction of higher-order nonlinear processes leading to stabilization (to a finite value) of the amplitude of the chaotic wave process. Mathematically, this is expressed in the synchronous “counteraction” of nonlinear processes of even and odd orders in the general vector-matrix model of an electrochemically active neuronal media being in a chaotic mode. It is noted that the state of hard self-excitation of nonlinear oscillations in an electrochemically active neuronal media leads to the appearance of a chaotic attractor in the state space. At the same time, the proposed vector-matrix model made it possible to find more general conditions for the appearance and evolution of chaotic wave processes in comparison with the initial Landau turbulence model and, as a result, to explain the occurrence of consistent nonlinear phenomena in an electrochemically active neuronal media.

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

Intravitreal use of vital dyes in combination with the action of endoillumination can induce a cyto- and phototoxic effect on posterior eye segment structures. The search for a staining agent with a maximum safety profile to retinal structures, intensively and selectively coloring vitreous body and vitreoretinal interface structure, remains relevant.Objective: to determine comparative viability of NIH / 3T3 mouse fibroblast cell culture with traditional agents for chromovitrectomy and “Vitreocontrast” suspension with and without endovitreal illumination.Materials and methods. NIH / 3T3 mouse fibroblast cultures contacted with agents for chromovitrectomy (MembraneBlue® Dual, Triamcinolone acetonide, “Vitreocontrast” suspension) and the corresponding controls in a volume of 50 μl / well. The test plate was irradiated with a Photon II illuminator (Synergetics, USA), working distance of 5 mm. The control tablet with the introduced preparations was not exposed to light. Next, the cells were washed and incubated, after which the morphology and lysis of the cells, as well as the number of proliferating relatively negative control of fibroblasts, were evaluated using the vital dye PrestoBlue Cell Viability Reagent. Negative control was the complete growth medium for the cultivation of mouse fibroblasts of the NIH / 3T3 line. The results of the cytotoxic reaction of a culture of mouse fibroblasts of the NIH / 3T3 line were interpreted using the table “The degree of cell response”.Results. Studies have shown that exposure to a source of endovitual illumination does not affect the cytotoxic effect of TA suspension and MembraneBlue® Dual dye. The TA suspension, both after light source and without it, has a moderate cytotoxic effect, and MembraneBlue® Dual has no cytotoxic effect on the culture of mouse fibroblasts of the NIH / 3T3 strain. Without light, “Vitreocontrast” suspension does not have cytotoxic effect on mouse fibroblasts culture NIH / 3T3 line. Light irradiation for 1 h increases the cytotoxicity of “Vitreocontrast” suspension to the level of unsharp cytotoxicity allowed by ISO Standard 10993-5-2011.Conclusion. The safety profile of MembraneBlue® Dual and “Vitreocontrast” suspension allows them to be recommended for use in endovitreal surgery. The cyto- and phototoxicity demonstrated in the experiment with TA suspension can reduce the functional outcomes of retinal surgery.

The OBJECTIVE was improving the results of diagnostics and assessment of the severity of patients with mechanical jaundice (MJ) at various stages of its development.METHODS AND MATERIALS. The basis of clinical and laboratory research was the data of 537 patients who were admitted during the period from 2010 to 2019. Principles of separation at the stage of the course of mechanical jaundice of non-tumor Genesis.RESULTS. Analysis of clinical and laboratory studies showed characteristic signs of various complications of cholelithiasis, accompanied by mechanical jaundice. Then, on this basis, specific symptoms characteristic of each stage of mechanical jaundice of non-tumor Genesis were determined.CONCLUSION. The course of mechanical jaundice, which develops with complications of cholelithiasis, has a phase-stage character, beginning with extrahepatic cholestasis, then-joining hepatocytolysis and ending with cholangitis. Initially, cholestasis and cytolysis are functional, which is confirmed by biochemical tests, so these processes are labile and reversible. This makes it possible to effectively use biliary decompression methods and infusion therapy with detoxification during treatment. Cholangitis is characterized by destructive morphological manifestations, so it is verified by specific clinical symptoms and laboratory tests characteristic of inflammatory-septic reactions and progresses to sepsis. Differentiation of stages of mechanical jaundice allows to personify surgical and conservative treatment of such patients.

At the moment, there are a huge number of risk factors for the development of a tumor process in the ovaries. One of the fundamental niches that allow us to suspect the presence of an imbalance in the body, and the presence of neoplasms is laboratory diagnostics. The purpose of our work was to evaluate changes in laboratory blood tests as a predictor of the development of oncological tumor process in the ovaries in patients of reproductive age (18-40 years). The collection and processing of laboratory data of 168 outpatient and inpatient medical histories of patients of reproductive age, operated on the basis of the Department of oncogynecology of the sbuz SOKOD for tumors and ovarian tumours from 2012 to 2015. Statistical analysis was performed using the statistical package SPSS21, license number 20130626-3 (An IBM Company; USA) and Microsoft Excel (Microsoft; USA). To achieve this goal, the patients were divided into groups according to the who morphological classification of 2013. A study of clinical blood analysis and biochemical blood analysis at the preoperative and early postoperative stages of treatment was conducted. The CA-125 titer was evaluated in two stages: as a qualitative and quantitative indicator. A study was also conducted for the presence of a genetic mutation in the BRCA 1 and 2 genes by taking a blood sample. In conclusion, we can conclude that on the basis of laboratory data it is impossible to conduct a differential diagnosis between tumors and tumor-like formations of the ovaries. When evaluating changes in General clinical blood tests, no significant changes were detected. Our work in a group of patients of reproductive age revealed the absence of mutations in the BRCA 1 and 2 genes. There was a significant increase in the tumor marker of the CA-125 titer (p = 0.015) in a group of patients with tumors and ovarian tumours aged 18–40 years.

Purpose. Estimation the local efficacy of the combined treatment — brachytherapy (BT) with laser coagulation (LC) in choroidal melanomas of juxta and parapapillary localization in the long-term follow-up. Patients and Methods. In 2009–2013, 50 patients with choroidal melanoma of juxta and parapapillary localization were examined and treated, 32 women and 18 men aged from 32 to 76 years old (average — 53.8 ± 9.6 years). The averaged tumor height was 3.8 ± 1.3 mm, basal diameter — 11.2 ± 2.4 mm. The combined organ-preserving treatment included a LC from the optic disc with subsequent BT was performed. The indications for this method were choroidal melanomas of juxta and parapapillary localization (the distance between the optic disc and the central border of the tumor was no more than 1.5 pd), with the absence of subretinal exudate and retinal detachment in this zone. The follow-up period after the combined treatment ranged from 18 to 102 months (Me = 60 months). Results. Complete tumor resorption was achieved in the majority — 38 (76.0 %) of cases, partial — in 11 (22.0 %) patients, stabilization of the process — only in one (2.0 %) patient, continued growth was not observed in any patient. The initial size of melanoma in patients with complete and partial tumor resorption showed similar averaged values, amounting to 3.8 ± 1.3 mm and 3.6 ± 1.1 mm (p > 0.05), respectively, basal diameter — 11.1 ± 2.4 mm and 11.4 ± 2.4 mm (p > 0.05), respectively. According to the duplex scanning, an increase in the distribution density of bloodflow in the projection of the tumor focus was noted compared with the initial data. In the spectral Doppler flow analysis study, an increase in the linear characteristics of the blood flow in the tumor’s own vessels after LC was recorded. Complications included optic neuropathy (88 %), hemorrhage (36 %) and secondary glaucoma (6 %). Conclusions. The combined treatment allowed to achieve high therapeutic results — 76.0 % of the total resorption of choroidal melanomas, which have unfavorable localization for BT. Given the main focus of local treatment of the choroidal melanoma on its destruction, this technique can be used to increase the effectiveness of the BT of choroidal melanomas of juxta and parapapillary localization.

Simpler than known structure of the system with error correction in calculations is proposed based on duplication and triplication of blocks with majority principle of choosing the values of signals. It is advisable to use the new fault-tolerant structure for automation devices with combinational logic. In fault-tolerant structure synthesis, the parity method is used to establish the fact of a fault in the main logic unit and the logical complement method is used determine incorrectly calculated output functions and to generate signals for their correction. The method also allows to adjust the values of incorrectly calculated functions. Structural diagram and description of error correction system are given. The synthesis algorithm of control equipment is described with minimization of the technical implementation complexity. The experiment results with control combinational circuits are given, confirming the high efficiency of proposed system structure with error correction.

У даній роботі у тривимірному евклідовому просторі E3 розглядаються загальні нескінченно малі (н.м.) деформації вищих порядків однозв'язних поверхонь, які мають важливе значення при вивченні їх неперервних деформацій. Завдання знаходження векторів зсуву цих деформацій зводиться до дослідження і розв'язку системи n рівнянь (або основних рівнянь) загальних н. м. деформацій скінченого порядку n, які отримані відносно довільно обраної на поверхні системи координат. Показано, що для замкнутих поверхонь додатньої гаусової кривини математичною моделлю цього завдання в сполучено-ізотермічній системі координат буде система n неоднорідних рівнянь комплексного виду, яка у випадку овалоїда приводиться до системи n інтегральних рівнянь. Використовуючи тензорні методи, апарат теорії узагальнених аналітичних функцій і методи функціонального аналізу, доведено, що регулярний овалоїд в E3 «в цілому» допускає загальну н.м. деформацію скінченого порядку n, яка однозначно визначається заздалегідь заданими 3n функціями. Знайдений їх геометричний зміст: завдання їх рівносильно завданням значень варіацій орта нормалі і елемента площі до порядку n включно. Векторні поля деформації при цьому визначаються з точністю до постійних векторів. Встановлено, що овалоїд буде жорстким щодо загальних н.м. деформацій скінченого порядку n тоді і тільки тоді, коли всі значення варіацій орта нормалі і елемента площі до порядку n включно тотожно рівні нулю. В якості прикладу поверхні, яка підтверджує отриманий результат, розглянута сфера радіуса R. Вектори зміщень при цьому знайдені в явному вигляді.

The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.

A brief overview of the possibility and readiness for digitalization of automotive industry companies has been presented. Since production processes and systems can be more intelligent, they allow to reach the high-quality standards required in service, what is the only way to maximize the improvement of its components from the very beginning of the process.Digitalization is one of the main levers to enhance business and increase competitiveness in the industry, that provides huge opportunities in the future. One of the important conditions is increase flexibility, that allows quickly to adapt production facilities and processes to changes in production and product lines of customer. The digitalization strategy should be applied transversely to all processes, starting from production, logistics, and quality till design, procurement, human resources, or finance. Transversal applying of digitalization strategy is possible in the creation of technology centers graphic engineering worldwide, the use of computerized simulation, CAD systems and virtual 3D environments for research, analysis, evaluation, validation and improvement of production processes. The important aspect to form the future is the development of innovative solutions that contribute turning new concepts into an industrial reality.The revolution in logistics digitalization gives automakers and service providers the opportunity to stay flexible and to be ahead of time. The main transformers are data and digital rethinking of the supply chain, that contribute a significant growth in trade. Advanced automakers create a digital workforce and program robots to solve repetitive tasks. All this provides full flexibility in software development with integrated in it robot’s hardware, fast iterations and adjustments, when it is necessary, which might not have been easily achieved using traditional methods. These projects can be extended to ones with greater investment. Mainly, more flexible and service-oriented solutions can be implemented as a response to speed and automation in logistics. Thereby, it is possible to satisfy the changing and demanding needs of customers now and in the future.

Introduction. Successful solution of the tasks of socio-economic development in the modern world is possible only in the case of effective functioning of the education system, which would provide training for a competitive personality, focused on continuous development and capable of “complying” with rapidly changing conditions of social reality. It is assumed that building an education development strategy, setting its goals and objectives should be based on the needs and problems of society, creating the basis for sustainable prosperity of the country and taking into account the needs of individual regions and territories. One of the most important mechanisms for the development of the regions of the Russian Federation remained the state educational order – officially developed task by the authorities for the creation of a specific educational product.The aims of the present article are the following: to determine the relationship between the educational state order, the volume of investments in the education system and the level of socio-economic condition of the regions; to identify the territories, which require support.Methodology and research methods. In the course of the study, the methods of comparative, statistical and system analysis, deduction and induction, generalisation and theoretical modelling were used. The object of the content analysis was the regional programmes for the development of education in 40 regions of the Russian Federation, covering the entire territory of Russia and reflecting the specific development of various regions depending on the remoteness of the capital, the status of the subject of the Russian Federation and the socio-economic development of the region.Results and scientific novelty. Russian regions were classified into four groups according to the results of the analysis of regional educational development – very high, high, moderate and low. A number of general and specific problems of education in the regions were identified: the lack of a single educational space and, as a result, the heterogeneity of educational programmes; regional leaders disregard for opportunities to expand socio-economic potential of the territories through the implementation of such programmes; the gap between the financing of education and its socio-economic importance, etc. The authors conclude that state educational order at the regional level is not considered as a vector of social and economic development of the territories, and its role is reduced to an educati onal level. It is established that the rate of development of education in the Russian regions does not always depend on their key socio-economic indicators. The greatest differentiation of the trend is recorded, when comparing the rate of development of the regional education systems and the gross regional product.Practical significance. The authors give recommendations to resolve disagreements over the formation of state education orders in the context of the socioeconomic development of regions. The need to develop regional economic development programmes, taking into account the specific social structure of the region, is emphasised. It is proposed to organise permanent social monitoring, which measures the quality of implementation of educational state order and strengthen the role of public participation when determining its content. The research materials and findings might be used by the executive authorities of the constituent entities of the Russian Federation, whose competence includes educational development issues.

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

Introduction. The paper is devoted to simulation modeling. Basic methods of the simulation mathematical modeling in the derivatives market are described. A group of realistic nonGaussian Levy processes that generalize the classical BlackScholes model is considered. The work objective is to study the most efficient methods of market forecasting, as well as the software implementation of the simulation mathematical modeling technique of the Russian derivatives market based on the Levy model. This research is relevant due to the demand for applications that simulate the dynamics of financial assets and evaluate options in realistic models of the derivatives market, allowing for jumps.Materials and Methods. Basic methods for forecasting the derivatives market, methods for determining the volatility rate at a known option price, are considered. The most effective types of Levy processes for the simulation mathematical modeling of the Russian derivatives market at the present stage are highlighted. The possibilities of the Java language for the implementation of mathematical methods are considered.Research Results. A program is developed in the Java programming language that implements the Levy mathematical model, which includes Gaussian and generalized Poisson processes. The program for calculating the mathematical method is created in the free integrated application development environment NetBeans IDE to work with any operating system.Discussion and Conclusions. The result of the simulation mathematical modeling analysis has shown that the most efficient methods in the derivatives market are those based on realistic non-Gaussian Levy processes. The software implementation of such mathematical methods can be used for educational purposes. The developed application has demonstrated high quality and speed of calculations using software resources.

This article describes a sporadic case of Brock’s bullous congenital ichthyosiform erythroderma. This is a rare hereditary disease from the group of genodermatoses with an autosomal dominant type of inheri - tance. The questions of the prevalence of dermatosis, the variability of the clinical picture, the timeliness of the clinical diagnosis are considered. Particular attention is paid to skin manifestations and their differential diagnosis.

calculated by the authors to use a simplified methodology that takes into account the indicators of average life expectancy at birth, the weighted average monthly wages and the average monthly pension, as well as the average duration of study and literacy of the population, for each of 54 municipalities and 8 urban districts of the Republic of Bashkortostan for the period of 2007 and 2013. A comprehensive study of spatial autocorrelation in the distribution of HDI in the republic was conducted in accordance with the five-step methodology proposed by the authors. At the first stage of the study, a weighted spatial matrix of inverse distances between the administrative centers of the municipalities was calculated. This matrix defined the spatial lag structure. At the second stage, which consisted in calculating the global and local indexes of spatial auto-correlation (Moran’s and Giris), the hypothesis about the presence of spatial autocorrelation in the HDI distribution was confirmed. Under the third stage, Moran’s scatterplots were used to visualize the spatial mutual influence of the HDI for specific municipalities for 2007 and 2013. The fourth stage consisted in spatial model estimation. Two specifications were considered: spatial auto-regression (SAR) and spatial error (SEM), both permitting to identify the mutual influence in the spatial distribution of the HDI in municipalities and urban districts. Coefficients of the models were estimated by using maximum likelihood approach. The final part of the study was devoted to the interpretation of the results of spatialregression modeling. R-Studio was used as a modelling tool.Results. It was shown that the distribution of the HDI in municipalities of the Republic of Bashkortostan is characterized by sustainable positive spatial auto-correlation. Moreover, we note an increase in dynamics of positive spatial correlation in the distribution of the HDI, which could be explained by the increasing role of urbanization and concentration of human resources in relatively large cities. There is even “a competitive struggle” going on in a number of municipalities for resources that contribute to raising the HDI. A number of municipalities form, however, a cluster of territories with a low level of human development. These areas are mainly located in the Northeast of the Republic. The estimation of spatial regression models allowed us overall to quantify the spatial auto-correlation dependencies in the distribution of human capital.Conclusion. The obtained results of spatial dependencies in the distribution of human capital can be used both in the development of strategies for the long-term socio-economic development of municipalities and serve as a basis for strategic planning of the development of the region.

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.

Aim: To evaluate an impact of a clinical decision support system (CDSS) in the clinical practice of a hospital on the quality and costs of treatment of patients with severe sepsis and septic shock.Materials and methods: We performed a retrospective analysis of the database in the medical information system qMS (SP.ARM, Russia, St. Petersburg) from 2015 to 2017 on 37,997 patients. In the first study period from January 2015 to June 2016, we analyzed the results of the conventional treatment regimen. From May to June 2016, the CDSS module was implemented into the qMS and the personnel was trained correspondingly. The data collected during the second study period mirrored the results of sepsis treatment with the use of CDSS. We assessed the average number of in-hospital days, duration of stay in the intensive care unit, number of septic shock cases, mortality, and treatment costs.Results: The diagnosis of sepsis was confirmed in 67 patients: in 1.4‰ (27/18,792) before the CDSS was implemented versus 2.1‰ (40/19,205) after the CDSS implementation (p < 0.01). It was found that the use of CDSS integrated into the hospital medical information system reduced the number of cases of septic shock development (p < 0.05). Lethality decreased by 10%. The sepsis-associated mortality showed a non-significant trend to decrease by 10% (p < 0.1). The implementation of the CDSS incorporated into the qMS helped to reduce the number of septic shock cases from 26% (7/27) to 7.5% (3/40) (p < 0.05). There was also a trend towards reduced duration of stay in intensive care unit, as well as towards decreased costs of sepsis treatment by 13% and efferent (extra-corporeal) treatments by 29%; however, the differences were not significant.Conclusion: The CDSS implementation for electronic monitoring of the patient's condition and changes in his/hers parameters allowed for an earlier diagnosis of sepsis. We identified some prerequisites for more rational utilization of medical resources, mainly due to early, targeted treatment of patients with severe sepsis and septic shock; however, additional studies are necessary.

We continue here our discussion of Part I, [18], by examining the local equivalence problem for partial differential equations and illustrating it with some examples, since almost any integration process or method is actually a local equivalence problem involving a suitable model. We terminate the discussion by inquiring on non-integrable Pfaffian systems and on their integral manifolds of maximal dimension.

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.

Introduction.The solution to the problem of transformation of phosphorus, nitrogen and silicon forms is studied. This problem arises under modeling phytoplankton dynamics in shallow-water bodies including the Azov Sea. The phytoplankton dynamics model is formulated as a boundary value problem for the system of diffusion-convection-response equations and takes into account the absorption and release of nutrients by phytoplankton, as well as the transition of nutrients from one compound to another. To calculate the initial conditions and parameters of the equations under which the steady-state regime occurs, the software is developed, which is based on the model describing changes in phytoplankton concentrations without considering current effects. This model is represented by a system of inhomogeneous differential equations. Based on the developed software, the initial conditions and parameters of the phytoplankton dynamics model in the Azov Sea are calculated experimentally.Materials and Methods.A 3D model of phytoplankton dynamics is considered taking into account the transformation of phosphorus, nitrogen and silicon compounds based on the system of nutrient transport equations. The case of a spatially uniform distribution of substances is considered to specify the parameters of the model at which the stationary modes occur. Because of simplification, a system of ordinary differential equations solved through the Runge-Kutta method is obtained.Research Results. The software is developed to specify the initial conditions and parameters of the phytoplankton dynamics model considering the transformation of phosphorus, nitrogen and silicon compounds. Several numerical experiments are performed under the assumption that the development of phytoplankton is limited by a single biogenic substance. As a result of the computational experiment, it can be seen that with the obtained values of the initial concentrations and parameters of the equations, stationary modes occur for the system of ordinary differential equations describing the case of the spatially uniform distribution of substances.Discussion and Conclusions.The mathematical model of the transformation of phosphorus, nitrogen and silicon forms in the problem of phytoplankton dynamics is studied. Stationary modes for the system of ordinary differential equations are obtained, for which the values of the system parameters and initial conditions are determined. The results obtained can be used in further simulation of the phytoplankton dynamics considering the transformation of phosphorus, nitrogen and silicon compounds with account for convection-diffusion, salinity, and temperature.

A technique is proposed for the experimental determination of internal friction in food materials, as well as a device based on a torsion pendulum with laser registration of the angle of rotation, which ensures the minimum relative displacement of the layers of the material under study in the measurement process. The definition of internal friction is based on finding the attenuation of a torsion pendulum connected to a cylinder immersed in the medium under study. As an example of the application of the method to food systems, the dependence of internal friction on viscosity in solutions of sucrose and glycerin is determined. This dependence is linear for sucrose solutions and non-linear for glycerol solutions, which may be due to the interaction of hydrated molecules in solutions. Examples of the application of internal friction measurements in the study of membrane concentration processes are given. The method of internal friction was used as an auxiliary rheological method in a comprehensive study of the process of deposition of concentrated substances on membranes during ultra- and nanofiltration of whey. Thus, it has been shown that, despite the thinness in comparison with the membrane, the polarization layer makes a significant contribution to the internal friction of the membrane-layer system. This leads to a sharp decrease in the flow of curd whey filtrate through the membrane, compared with the flow of water under the same conditions. The dependence of internal friction on the concentration in the curd whey that underwent electroflotation treatment was determined, as well as the thermal isomerization of the lactose contained in it. Using such curves, the relationships between the permeability coefficient Lp * in the Kedem–Kachelsky equation and the parameters of the nanofiltration process of given food systems can be found.

В статті дослiджується структура матриць над областю головних iдеалiв вiдносно перетворення подiбностi. В другому розділі наведено допоміжні результати. В цьому розділі вказано трикутну формуматрицi відносно перетворення подібності, мінімальний многочлен якої розкладається в добуток різних лінійних множників. В розділі 3 доведено, що форма Хессенберга матриці A з незвідним мінімальним квадратичним многочленом m(λ) є блочно-трикутна матриця з блоками вимірності 2х2 на головній діагоналі та з характеристичними многочленами m(λ). У четвертому розділі доведено, що матриця A із мінімальним многочленом m (λ) = (λ-α) (λ-β), α ≠ β подібна нижній блочно-трикутній матриці, діагональними блоками якої є діагональні матриці з елементами α i β на головних діагоналях відповідно. Як наслідок вказано канонічну форму інволютивної матриці над кільцем цілих чисел відносно перетворень подібності.

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

On the example of data obtained during psychological and psychiatric examination and further rehabilitation of women and adolescent girls victims of crimes against sexual inviolability of the person, the article describes mental and behavioral disorders developed as a result of psychotrauma. Reflected features of the posttraumatic mental and behavioral disorders course and differences in manifestations in the age aspect. It has been established that the clinical picture of the course of the disease depends on such factors as the age, the psychological maturity of the victim of violence, the severity of committed violence. Also, the analysis of the work allowed us to identify a number of negative phenomena that contributed to the protracted, undulating course of reactive states. These include a high degree of stigmatization (self-stigmatization) in society about sexual offenses and, as a result, bullying victims of sexual violence, as well as the need to participate in forensic investigative events. It is shown that the frequency of face-to-face interrogation and the duration of the trial affect the severity and duration of the mental consequences of the psychological trauma.

Purpose: to evaluate the efficiency of vitreous floaters removal using the Nd: YAG-laser DIXION LPULSA SYL-9000 with a coaxial light splitter. Patients and Methods. The study involved 24 patients (24 eyes) with Weiss ring-type floating vitreous opacities. All patients underwent laser vitreolysis on a Nd: YAG-laser DIXION LPULSA SYL-9000 with a coaxial light splitter. Laser procedures were performed at OPTIMED Laser Recovery Center. The age of the patients was from 47 to 81 years (average 58.3 ± 8.9 years). Among the patients 15 (62.5 %) were women and 9 (37.5 %) — men. Before procedure, a complete ophthalmological examination including registration of best corrected visual acuity (BCVA), autorefractometry, pneumotonometry, biomicroscopy, fundus examination, ultrasonic biometry, ultrasound B-scan was performed. Statistical processing was performed using Statistica v.10.0.0 (StatSoft®, Inc.) and Excel (Microsoft Office Professional 2016) software. The patients were observed for three months before the procedure. Follow-up examination was performed one day after the procedure. Results. After laser vitreolysis procedure all patients had a subjective vision quality improvement. In 18 (75 %) patients, floating “flies” completely disappeared. The remaining patients noted the disappearance of a large floating spot. Conclusion. The laser vitreolysis on the Nd: YAG-laser with a coaxial light splitter allows to improve the quality and visual acuity in patients with vitreous floater. The obtained results demonstrated that Nd: YAG-laser DIXION LPULSA SYL-9000 is effective and could be recommended for treatment of symptomatic vitreous floaters.

In this paper we introduce a new class of foliations on Rie-mannian 3-manifolds, called B-foliations, generalizing the class of foliations of non-negative curvature. The leaves of B-foliations have bounded total absolute curvature in the induced Riemannian metric. We describe several topological and geometric properties of B-foliations and the structure of closed oriented 3-dimensional manifolds admitting B-foliations with non-positive curvature of leaves.

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by $\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$ induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)= \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{O}_{f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this situation $\Gamma_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$ consists of more than one point., Comment: 5 pages

Block ciphers form one of the main classes of cryptographic algorithms. One of the challenges in development of block ciphers, like any other cryptographic algorithms, is the analysis of their cryptographic security. In the course of such analysis, statistical testing of block ciphers is often used. The paper reviews literature on statistical testing of block ciphers.The first section of the paper briefly and informally discusses approaches to the definition of the concept of a random sequence, including the Kolmogorov, von Mises, and Martin-Löf approaches and the unpredictability-related approach. However, all these approaches to the definition of randomness are not directly applicable in practice.The second section describes statistical tests of binary sequences. It provides brief descriptions of the tests included in the DieHard, NIST STS, RaBiGeTe statistical test suites.The third section provides the appropriate information to present further the operation modes of block ciphers.The fourth section deals with techniques for statistical testing of block ciphers. Usually such techniques lie in the fact that based on the block cipher under test, various generators of the pseudorandom sequences are built, with their output sequences being tested using any suite of statistical tests. The approaches to the construction of such generators are given.The paper describes the most known statistical test technique for block ciphers among the submitted for the AES competition. It is a technique the NIST uses for statistical testing of ciphers. In addition, there are other techniques mentioned in the literature.In conclusion the paper states that there is a need to develop new techniques for statistical testing of block ciphers.The paper support was provided from the Russian Foundation for Basic Research in the framework of the research project No. 16-07-00542 supported

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

One of the ways of knowing the events of military history is to reproduce them using mathematical models. Based on the analysis of the fighting operations of the 4th Panzer Brigade of the Red Army in the vicinity of the city of Mtsensk in early October 1941, the capability to provide mathematical modeling of the fragments of these combat operations and the application of the apparatus of Markov random processes for these purposes is substantiated.The effectiveness of tanks depends not only on their technical properties, but also on the ways they are used on the battlefield. At the same time, combat effectiveness of tanks is commonly understood as their effectiveness in conditions when the methods of conducting combat operations by each of the opposing sides are the best.The battle outcome is probabilistic. It has certain regularity, depending on the combat tactics. The battle can be imagined as a multitude of randomly dueling fights between tanks, differing in their location and range of fire. A study of the probability of a system transition from each transient state to the next leads to the construction of mathematical models that allow calculating the ratio of losses of opposing sides.Based on the facts of military history and discovered regularities, the mathematical models are constructed to allow reproducing various fragments of combat according to the scheme of the Markov random process, and on their basis calculations are performed. The dependence of the ratio of the losses of the opposing sides depending on the number of firing positions used by the ambush tanks was established, provided that the change of these positions was made imperceptibly for the enemy.The obtained results can be used to develop tactical methods of using tanks in antiterrorist operations.

Choroidal melanoma is a primary malignant intraocular tumor with unfavorable vital prognosis. Factors predisposing to invasion of choroidal melanoma into the optic nerve include its juxtapapillary localization, the large tumor size, its diffuse growth, the presence of necrosis in the tumor, increased intraocular pressure and the epithelioid cell type of the tumor. The authors describe the clinical case of a 31-year-old patient with a primary diagnosis of an optic nerve neoplasm. Peripapillary edema and indistinct optic disc margins were found by ophthalmoscopy. There was a slightly protruding gray-slate lesion with indistinct boundaries near the optic nerve disc. The retina was edematous with multiple merging hemorrhages. The data obtained by echography (the focus protruding both inside the eye and along the optic nerve trunk), color Doppler mapping (single own vessels in the projection of the orbital part of the mass) and computed tomography allowed for the suspicion of the primary choroidal melanoma. To clarify the nature of the process, a trans-conjunctival orbitotomy with a revision of the orbit and the optic nerve was performed. The expansion of the proximal part of the optic nerve with partial breakthrough of the optic nerve shells was revealed intraoperatively. The patient underwent enucleation with morphological and molecular genetic verification of the primary choroidal melanoma with invasion of the optic nerve. The data obtained (absence of chromosome 3 monosomy, deletion of the short arm of chromosome 1, methylation of promoter regions of the RASSF1A gene, mutations in the 4 and 5 exons of the GNAQ gene, CT genotype of the polymorphic marker C3435T of the ABCB1 gene) indicate a relatively favorable vital prognosis, despite the extrascleral growth of the tumor and its mixed cell type. The patient underwent a course of anti-relapse proton irradiation of the orbit with a total dose of 58.8 Gy at 80–90% isodose. Currently, 7 years after the treatment, the patient continues to be followed up by ophthalmologists and oncologists, without any signs of local recurrence and distant metastases. This clinical example demonstrates the possibility of extraocular extension of a small intraocular choroidal melanoma simulating an optic nerve tumor, and confirms the need for a thorough clinical and instrumental examination for a correct diagnosis and adequate management of the patient.

Using a previously derived kinetic model for an imperfect gas, we have derived an equation of state for a gas in a thermodynamic equilibrium and an expression for the critical temperature as a function of an interaction potential between molecules.

We consider a numerically stable direct multiplicative algorithm of solving linear equations systems, which takes into account the sparseness of matrices presented in a packed form. The advantage of the algorithm is the ability to minimize the filling of the main rows of multipliers without losing the accuracy of the results. Moreover, changes in the position of the next processed row of the matrix are not made, what allows using static data storage formats. Linear system solving by a direct multiplicative algorithm is, like the solving with $LU$-decomposition, just another scheme of the Gaussian elimination method implementation. In this paper, this algorithm is the basis for solving the following problems: Problem 1. Setting the descent direction in Newtonian methods of unconditional optimization by integrating one of the known techniques of constructing an essentially positive definite matrix. This approach allows us to weaken or remove additional specific difficulties caused by the need to solve large equation systems with sparse matrices presented in a packed form. Problem 2. Construction of a new mathematical formulation of the problem of quadratic programming and a new form of specifying necessary and sufficient optimality conditions. They are quite simple and can be used to construct mathematical programming methods, for example, to find the minimum of a quadratic function on a polyhedral set of constraints, based on solving linear equations systems, which dimension is not higher than the number of variables of the objective function. Problem 3. Construction of a continuous analogue of the problem of minimizing a real quadratic polynomial in Boolean variables and a new form of defining necessary and sufficient conditions of optimality for the development of methods for solving them in polynomial time. As a result, the original problem is reduced to the problem of finding the minimum distance between the origin and the angular point of a convex polyhedron, which is a perturbation of the $n$-dimensional cube and is described by a system of double linear inequalities with an upper triangular matrix of coefficients with units on the main diagonal. Only two faces are subject to investigation, one of which or both contains the vertices closest to the origin. To calculate them, it is sufficient to solve $4n - 4$ linear equations systems and choose among them all the nearest equidistant vertices in polynomial time. The problem of minimizing a quadratic polynomial is $NP$-hard, since an $NP$-hard problem about a vertex covering for an arbitrary graph comes down to it. It follows therefrom that $P = NP$, which is based on the development beyond the limits of integer optimization methods.

We study a mixed problem for systems of semilinear hyperbolic equations with nonlinear dissipation and focusing, as well as with ‘defocusing’ sources. Using the combined method of regularization and the Galerkin method, the theorem of local solvability is proved. For systems with a focusing source, an a priori estimate is obtained, from which follows the corresponding theorem on global solvability. In the case of systems of hyperbolic equations with a ‘defocusing’ source, it is proven that if the order of growth of the variables does not exceed the order of growth of the dissipative term, then there is global solvability.

The problem of determining the stress-strain state a sandwich plate with a transversally soft core in one-dimensional geometrically nonlinear formulation has been considered. It has been assumed that in the right end section the edges of the carrier layers are rigidly clamped and there is no adhesive bond between the core and the support element, and in the left end section the edges of the carrier layers of the plate are hinged on an absolutely stiff in the transverse direction diaphragm glued to the end section of the core, the load is applied in the middle surface of the first carrier layer from the left side. On the basis of the generalized Lagrange principle, a general formulation has been created in the form of an operator equation in the Sobolev space. The properties of the operator have been established, i.e., its pseudo-monotonicity and coercivity.Thus, a theorem on the existence of a solution has been established. A two-layer iterative method has been suggested for solving the problem. Considering additional properties of the operator, i.e., its quasipotentiality and bounded Lipschitz continuity, convergence of the method has been investigated. The variation limits of the iteration parameter ensuring the method convergence have been established. A software package has been developed, with the help of which subsequent numerical experiments for the model problem of longitudinal-transverse bending of a sandwich plate have been carried out. Tabulation of both longitudinal and transverse loads has been carried out. The obtained results show that, in terms of weight perfection, the sandwich plate of an asymmetric structure with unequal thicknesses of the carrier layers is the most rational and equally stressed one under the considered loading.

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\to\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\mathbb{R}^2\to\mathbb{R}$ without multiple factors. Together with previous results of the second author this allows to compute similar groups for certain classes of smooth functions $f:N\to\mathbb{R}$ on non-orientable compact surfaces $N$.

It is known that the most important affinor structures on smooth manifolds are almost complex structures, almost product structures, f-structures of Kentaro Yano, and some others. However, a new type of affinor structures was introduced and intensively discussed in differential geometry in the last decade. These are the so-called Golden structures first introduced by M. Crasmareanu and C.-E. Hretcanu using the well-known quadratic equation for the Golden ratio. Since then, a number of papers have been devoted to the study of the integrability of Golden structures, compatible Riemannian metrics and connections, submanifolds in such manifolds, etc. At the same time, invariant Golden structures on homogeneous manifolds have not appeared in these investigations. In this paper, a wide collection of invariant Golden structures on homogeneous generalized symmetric spaces is presented. More precisely, we obtained a complete description of all canonical Golden structures on homogeneous k-symmetric spaces. A remarkable feature of these structures is that all of them are invariant with respect to both the acting Lie group as well as the generalized symmetries of order k of homogeneous k-symmetric spaces., Известно, что наиболее важными аффинорными структурами на гладких многообразиях являются почти комплексные структуры, структуры почти произведения, f-структуры Кентаро Яно и некоторые другие. Однако в последнее десятилетие новый тип аффинорных структур был введен и интенсивно обсуждался в дифференциальной геометрии. Это так называемые золотые структуры, впервые введенные М. Красмареану и К.-Э. Хретчану с использованием хорошо известного квадратного уравнения для золотого сечения. С тех пор ряд работ был посвящен изучению интегрируемости золотых структур, согласованных римановых метрик и связностей, подмногообразий в таких многообразиях и т.д. В то же время инвариантные золотые структуры на однородных многообразиях в этих исследованиях не появлялись. В данной статье предъявлен обширный класс инвариантных золотых структур на однородных обобщенных симметрических пространствах. Более точно, мы получили полное описание всех канонических золотых структур на однородных k-симметричных пространствах. Примечательной особенностью этих структур является то, что все они инвариантны как относительно действующей группы Ли, так и относительно обобщенных симметрий порядка k однородных k-симметрических пространств.

Последнее время становится актуальным изучение (псевдо)римановых многообразий с различными метрическими связностями, отличными от связности Леви-Чивита.Метрическая связность с векторным кручением (также известная как полусимметрическая связность) является одной из часто рассматриваемых связностей. Связь между конформными деформациями римановых многообразий и метрическими связностями с векторным кручением на них была установлена в работах К. Яно.А именно: риманово многообразие допускает метрическую связность с векторным кручением, тензор кривизны которой равен нулю, тогда и только тогда, когда оно является конформно плоским.Кроме того, данная связность играет важную роль в случае двумерных поверхностей, так как в этом случае любая метрическая связность является связностью с векторным кручением.Таким образом, возникает задача об изучении (псевдо)римановых многообразий с метрической связностью с векторным кручением, тензор кривизны которых равен нулю.Данная работа посвящена решению поставленной задачи в случае трехмерных метрических групп Ли. Кроме того, приводится математическая модель, позволяющая вычислять компоненты тензора кривизны метрической связности с векторным кручением в случае метрических групп Ли., Recently, the study of (pseudo)Riemannian manifolds with different metric connections different from the Levi-Civita connection becomes relevant.A metric connection with vectorial torsion (also known as a semi-symmetric connection) is one of the often considered connections.The connection between the conformal deformations of Riemannian manifolds and metric connections with vectorial torsion on them was established in the works of K. Yano.Namely, a Riemannian manifold admits a metric connection with vectorial torsion, the curvature tensor of which is zero, if and only if it is conformally flat.Moreover, this connection plays an important role in the case of two-dimensional surfaces since, in this case, any metric connection is a connection with vectorial torsion.Thus, the problem of studying (pseudo)Riemannian manifolds with metric connection with vectorial torsion, the curvature tensor of which is zero, is arisen.This paper is devoted to solving the problem in the case of three-dimensional metric Lie groups. In addition, a mathematical model is presented that allows one to calculate the components of the curvature tensor of a metric connection with vectorial torsion in the case of metric Lie groups.

Darboux theorem to more general context of Frechet manifolds we face an obstacle: in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$ is the standard symplectic form on $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $ generated by a time-dependent vector field $ X_t $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless, it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the classifying space does not depend on the point of the manifold and a suitable smoothness condition holds. If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar. In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields we prove the Darboux theorem.