The article is dedicated to the memory and scientific legacy of the prominent Ukrainian mathematician Heorhii Mykolajovych Polozhii (1914 1968). The main ideas of the theory of generalized analytic functions, the summary representation method are described and some new application results in mathematical modelling of nonlinear quasi-ideal processes in LEF layers are considered.

The paper proposes an approach for finding the optimal position of sources of known power for the quasi-linear Richards equation in a rectangular area. The Kirchhoff transformation is applied with the introduced scaling of coordinates and powers of submerged sources, which allows formulating a dimensionless problem. The task of this study is to find the position of submerged sources - such that the distribution of moisture at the final moment of time is close to the given values or the given target function.

The review describes the main stages of formation and development of the Kyiv School of Mathematical Theory of Filtration. This review focuses on the scientific ideas and results of its leader, a prominent Ukrainian scientist, Academician of the National Academy of Sciences of Ukraine Ivan Ivanovych Lyashko. The journal “Cybernetics and Systems Analysis” systematically publishes the works of I. I. Lyashko’s students, in which his ideas and achievements are further developed.

The paper describes novel machine-learning, high-precision methods for diagnosing breast cancer based on the fractal dimension of buccal epithelium nuclei. The method uses the Random Forest algorithm and logistic regression applied to data on fractal dimension of images estimated by blue, green, and red channels. We investigated the control group (29 woman), patients with breast cancer at the second stage (68 woman), and patients with fibroadenomatosis (33 woman). The dataset consists of 20256 photos of interphase nuclei of buccal epithelium (6752 nuclei photographed without filter, through a yellow filter, and through a violet filter). All images were preprocessed using tools of morphological analysis. The best results are obtained when using all data channels: 95% accuracy, 92% sensitivity, and 100% specificity. The fractal dimension is estimated using the Hurst exponent. High performance and non-invasiveness of the method gives the hope that it well be useful for clinical applications.

The gold standard for managing muscle-invasive bladder cancer is radical cystectomy (RCE). The RCE is a treatment, which carries high burden of perioperative morbidity and mortality. As biomolecular markers make muscle-invasive high-grade bladder cancer (HGBC) an entity different from non-invasive papillary disease, we tested a hypothesis that alternative bladder preserving surgery (BPS) approaches, such as partial cystectomy and transurethral resection of the bladder would not compromise the oncological results of treating HGBC in selected patients. Aim: To study the cancer specific survival of HGBC patients depending on the mode of surgical treatment - RCE, partial en-block cystectomy, and transurethral resection of the bladder in the practice of the Departments of Urology and Oncology of Bogomolets National Medical University, and to assess the prevalence of bladder sparing surgical management of HGBC in local practice as a part of trimodal treatment approach to bladder preservation. Materials and Methods: Retrospectively we studied the medical records of 3597 urothelial bladder cancer patients, of whom 346 (10%) had high-grade disease and who underwent surgical treatment in 2004-2017. All patients were studied with contact computed tomography of the chest, abdomen, pelvis, and biopsy of the tumor. Based on the results of the diagnostic workup the choice of surgical treatment between RCE, partial cystectomy and transurethral resection was made considering the size of the tumor, location of the tumor in the bladder in relation to the bladder neck, and technical and oncological feasibility of performing the bladder sparing surgery. Kaplan - Meier survival curves were built to compare the results of survival per cancer stage and type of surgical treatment. Survival data of the patients were collected from the cancer registry maintained at the Kyiv Municipal Clinical Oncological Center. Results of data analysis were controlled for confounding parameters, such as adjuvant treatment: perioperative radiotherapy, and chemotherapy. Results: Median follow-up was 93 months (1-226 months). Males were 276 (80%). Average age at diagnosis was 62 ± 4.5 years. By the time of the study 61% of patients have died due to the progression of the disease. All patients with stage I disease (7% or 24 patients) were managed with bladder-sparing surgery. In muscle-invasive disease (309 patients), the RCE was performed in 109 (35.3%) patients, partial cystectomy was performed in 79 (25.6%) patients, and transurethral resection - in 121 (39.1%) patients. The overall 5-year survival of HGBC patients after radical surgical treatment (RCE/BPS) for stage I patients was 0%/83%, for stage II - 43%/58%, for stage III - 37%/42%, and for stage IV - 10%/40%. A total of 44 patients (12.7% of all treated, and 19.6% of treated with bladder sparing) received postoperative radiotherapy after bladder-sparing surgery. A total of 14 patients (4% of all treated) received postoperative chemotherapy. Conclusion: Bladder sparing surgery (partial en-block cystectomy, and transurethral resection of the bladder) in selected patients is not inferior to RCE in terms of cancer-specific survival when treating patients with HGBC of all stages. The bladder sparing surgery was performed in 64.7% of patients with high grade bladder cancer. Utilization of adjuvant treatment is low, 12.7% for postoperative radiotherapy, and 4% for perioperative chemotherapy.

The authors propose an algorithm for finding the optimal source capacity for the two-dimensional quasi-linear Richards equation in a rectangular domain. The Kirchhoff transformation with scaling of coordinates and capacities of buried sources is used, which allows formulating a dimensionless problem. The existence of a solution to the problem of optimizing moisture transfer in an unsaturated porous medium is substantiated. The task of this study is to find the capacity of sources buried in a porous medium, such that the humidity distribution at the final instant of time is close to the given indicators or the objective function. The numerical solution approximates the optimal values of the sources.

The COVID-19 pandemic has exposed an urgent problem of forecasting the epidemic curves. Now, the most popular tools for outbreak forecasting are SIR and SEIR models. The main drawback of these models is that they use uncertain parameters that may be invalid. We can solve this problem using machine learning models trained on real data. Every such model has its own accuracy. Yet, a comparison of the accuracy of these models is quit a difficult task due to their statistical nature. Two models may have different but statistically equivalent errors. Thus, we may reduce the comparison of forecasting models to testing the homogeneity of samples of their errors. In this chapter, we represent a novel nonparametric test of sample homogeneity and its application to the comparison of five forecasting models for the COVID-19 epidemic curve. Also, we show the efficiency of the proposed method. This test has a guaranteed significance level due to using an exact confidence interval for the binomial proportion.

The article is dedicated to several gradient based methods for solving a two-dimensional humidification problem, described by Richards equation. Several assumptions are made: water is assumed incompressible, external pressure and temperature are constant. The initial state and desired function are known, while the optimal source power should be calculated. Kirchhoff transformation is applied to the initial equation to simplify the stated problem. Time and space coordinates are scaled to get linear dimensionless equation, which can be easily discretized over space and time. Numerical methods are applied to rewrite and solve the system. Also gradient methods are applied for cases, where it is possible to define the optimization functional for every allowed source power.

The article describes a numerical method for optimizing the chemotherapy of malignant tumors on the basis of drug delivery using increased convection. The problem of optimal control with point sources for reaching the desired intratumor distribution of drugs in macroscopic scale granting the properties of intersticial space and effects of convective diffusion is considered. The efficiency of proposed algorithm for optimal control is shown.

The paper describes new simple non-parametric tests for recognizing change points in a time series separating fragments obeying different distributions. The test is based on Hill’s assumption A(n) and the generalized scheme of the Bernoulli scheme. To identify the change point, it is proposed to use nonparametric tests that use the effect of stabilizing the relative frequency of success in the Bernoulli scheme. Both samples without repeated elements and samples with repeated element are allowed. The test allows both online and offline implementations without requiring significant computing power. Experiments demonstrate the stability and sensitivity of the proposed tests. The tests recognize change points between two distributions with high accuracy, both in the case of different location parameters and in the case of different scale parameters.

In this paper a one-dimensional nonlinear Richards equation describing fluid flow in porous medium with inserted equalpowered sources is studied. An experimental iterational method is proposed to find source power to minimize the deviation of received humidity values from target values. Modeling was performed using numerical difference approximation of derivatives, resulting into a system of nonlinear equations with dependence from previous time step. The offered method allows to perform modeling for different source power values, and chooses the most suitable one.Iterations stop when they reach average modular difference value less than calculation error of numerical difference scheme. Here explicit scheme was used to save time, equations were tested for unsaturated medium only to avoid flooding the area, so source power is tested with given limitations. Results of simulations and choice for next source power approximations are described and compared until solution is found. This approach is considered as experimental so we plan to perform more analysis in the future.

The article is dedicated to a two-dimensional humidification problem, described by Richards-Klute equation. Several simplifications are made: water is assumed incompressible, external pressure and temperature are constant. The initial state and desired function are known, while the optimal source power should be calculated. The initial equation is transformed using the Kirchhoff transformation. New scaled values are introduced to get linear dimensionless equation, which can be easily discretized over space and time. Then implicit (or explicit) methods and numerical methods are applied to solve the system. In general, the problem can also take into account heat transfer, chemical reactions or surface heterogeneity leading to more complicated equations, so the research will continue.

The paper describes a nonparametric test for recognizing the point of change in the time series, before and after which the values of the time series obey different distributions. The test is based on the Matveychuk–Petunin scheme, which is a generalization of the Bernoulli scheme using Dempster–Hill procedure. To recognize the point of change in the time series, a simplified Klyushin–Petunin homogeneity criterion is used, based on an exact confidence interval. The test works equally well with samples that do not have ties, as well as with samples having ties. It allows both online and offline implementations. The test compares segments of time series with high accuracy with a significance level of no more than 0.05. The sensitivity and stability of the proposed test is higher than that of its classical counterparts. The test provides high accuracy of recognition of two heterogeneous random samples for both the location shift hypothesis and the scale shift hypothesis. The proposed approach has wide practical applications in all areas where time series arise.

The paper presents the results of the comparison of two nonparametric methods of authorship identification of the Ukrainian literature texts. The paper describes the implementation of the corresponding methods based on the Klyushin–Petunin tests and its simplified version. The method of n-gram selection is applied. For testing a collection of texts up to 200,000 characters from 10 authors was used. As a result of carrying out the test, it was found out that the simplified test appears to be more sensitive and specific, and monograms and bigrams in opposite to trigrams provide clear detection of authorship.