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The paper considers integro-differential equations of fractional order moisture transfer with the Bessel operator. The studied equations contain the Bessel operator, two Gerasimov – Caputo fractional differentiation operators with different orders $\alpha$ and $\beta$. Two types of integro-differential equations are considered: in the first case, the equation contains a non-local source, i.e. the integral of the unknown function over the integration variable $x$, and in the second case, the integral over the time variable τ, denoting the memory effect. Similar problems arise in the study of processes with prehistory. To solve differential problems for different ratios of $\alpha$ and $\beta$, a priori estimates in differential form are obtained, from which the uniqueness and stability of the solution with respect to the right-hand side and initial data follow. For the approximate solution of the problems posed, difference schemes are constructed with the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ for $\alpha\neq\beta$. The study of the uniqueness, stability and convergence of the solution is carried out using the method of energy inequalities. A priori estimates for solutions of difference problems are obtained for different ratios of $\alpha$ and $\beta$, from which the uniqueness and stability follow, as well as the convergence of the solution of the difference scheme to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

We examine a phenomenological algorithm for generating morphology of astrocytes, a major class of glial brain cells, based on morphometric data of rat brain protoplasmic astrocytes and observations of general cell development trends in vivo, based on current literature. We adapted the Space Colonization Algorithm (SCA) for procedural generation of astrocytic morphology from scratch. Attractor points used in generation were spatially distributed in the model volume according to the synapse distribution density in the rat hippocampus tissue during the first week of postnatal brain development. We analyzed and compared astrocytic morphology reconstructions at different brain development stages using morphometry estimation techniques such as Sholl analysis, number of bifurcations, number of terminals, total tree length, and maximum branching order. Using morphometric data from protoplasmic astrocytes of rats at different ages, we selected the necessary generation parameters to obtain the most realistic three-dimensional cell morphology models. We demonstrate that our proposed algorithm allows not only to obtain individual cell geometry but also recreate the phenomenon of tiling domain organization in the cell populations. In our algorithm tiling emerges due to the cell competition for territory and the assignment of unique attractor points to their processes, which then become unavailable to other cells and their processes. We further extend the original algorithm by splitting morphology generation in two phases, thereby simulating astrocyte tree structure development during the first and third-fourth weeks of rat postnatal brain development: rapid space exploration at the first stage and extensive branching at the second stage. To this end, we introduce two attractor types to separate two different growth strategies in time. We hypothesize that the extended algorithm with dynamic attractor generation can explain the formation process of fine astrocyte cell structures and maturation of astrocytic arborizations.

The article considers the mathematical aspects of the problem of identifying a multicomponent scattering medium based on pulsed multienergy X-ray irradiation data. X-ray diagnostics problems are of considerable interest from both theoretical and practical points of view, and radiographic methods are indispensable in non-destructive testing of products.

Within the framework of a mathematical model based on a non-stationary integro-differential equation of radiation transfer, the inverse problem of finding the attenuation coefficient for radiation known at the boundary of the region and the problem of identifying a substance based on the found values of the attenuation coefficient on a discrete set of irradiation energies of the medium are formulated.

A preliminary processing of a wide list of substances of interest in computed tomography was carried out to determine the possibility of their identification by an approximately specified radiation attenuation coefficient characterizing the medium. When analyzing the degree of proximity of substances in a certain norm, it was found that the set of all possible substances potentially contained in the medium is divided into a finite number of non-intersecting clusters. For a sufficiently short duration of the probing signal, the scattering component of the radiation leaving the medium is asymptotically small. This circumstance allows us to reduce the inverse problem for the radiation transfer equation to the problem of inverting the Radon transform from the attenuation coefficient. The possibility of unambiguous or partial identification of a substance by varying the duration of the probing pulse and the number of energy levels of irradiation of the medium is analyzed using numerical modeling methods on a specially developed digital phantom.

The paper presents a hybrid artificial neural network model based on an architecture that incorporates a convolutional image encoder (CNN) and an attention module (Attention-based Multiple Instance Learning, Attention MIL). This module aggregates informative features from a sequence of frames capturing the flame spraying process. Additional technological parameters—air pressure, propane pressure, and standoff distance — are integrated into the model via a tabular channel, enabling it to account for the relationship between visual data and numerical process regime characteristics. The software implementation was developed using the Streamlit platform and the PyTorch library. It features an interactive interface for model training and result visualization, analysis of attention weights across frames, and a prediction mode for output characteristics: surface roughness ($R_a$) and the mass of the deposited coating ($m$). Experimental studies were conducted on data from real-world technological processes, and a comparative analysis of the accuracy of various model configurations was performed. The results demonstrate that the hybrid neural network, which combines visual and tabular features, achieves higher prediction accuracy compared to models using only a single modality. Furthermore, when comparing different implementations of the hybrid network, it was established that using the attention mechanism to process the series of flame spray images provides a significant increase in accuracy over a simple averaging of features without attention. The application includes an attention visualization module that creates a montage of the most significant frames and displays their attention weights, allowing users to identify which frames had the greatest influence on the prediction. The model’s capability for export to the ONNX format for integration into process control systems is also demonstrated. The proposed approach showcases the effectiveness of fusing visual and tabular information for manufacturing process monitoring tasks. The model can serve as a foundation for developing a decision support system or an automated quality control system for coatings produced by flame spraying. The limitations of the implemented model and prospects for its further development are also considered.

The article is dedicated to mathematic modeling of natural water desalination process by method of thermal distillation. The article gives the equations which allow describing the processes of film flowing and boiling of water, steam condensation and vacuum maintenance. The article presents the algorithm of calculation, implemented in MatLab computer mathematic system and Excel electronic tables, and the initial data required for the calculation. The model has been checked for adequacy. The calculation of ten-effect distillation system is given. The results of work can be used in design and optimization of process conditions for distillation systems.

The problem of accumulation and the statistical analysis of computer experiment results are solved. The main experiment program is considered as the data source. The results of main experiment are collected on specially prepared sheet Excel with pre-organized structure for the accumulation, statistical processing and visualization of the data. The created method and the program are used at efficiency research of the scientific researches which are carried out by authors.

WS-BPEL is a widely accepted standard for specification of business distributed and parallel processes. This standard is a mismatch of algebraic and Petri net paradigms. Following that, it is easy to specify WS-BPEL business process with unwanted features. That is why the verification of WS-BPEL business processes is very important. The intent of this paper is to show some possibilities for conversion of a WS-BPEL processes into more formal specifications that can be verified. CSP and Z-notation are used as formal models. Z-notation is useful for specification of abstract data types. Web services can be viewed as a kind of abstract data types.

The repressor is the first genetic regulatory network in synthetic biology, which was artificially constructed in 2000. It is a closed network of three genetic elements — $lacI$, $\lambda cI$ and $tetR$, — which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In this paper, the nonlinear dynamics of a modified repressilator, which has time delays in all parts of the regulatory network, has been studied for the first time. Delay can be both natural, i.e. arises during the transcription/translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using synthetic biology technologies. It is assumed that the regulation is carried out by proteins being in a dimeric form. The considered repressilator has two more important modifications: the location on the same plasmid of the gene $gfp$, which codes for the fluorescent protein, and also the presence in the system of a DNA sponge. In the paper, the nonlinear dynamics has been considered within the framework of the deterministic description. By applying the method of decomposition into fast and slow motions, the set of nonlinear differential equations with delay on a slow manifold has been obtained. It is shown that there exists a single equilibrium state which loses its stability in an oscillatory manner at certain values of the control parameters. For a symmetric repressilator, in which all three genes are identical, an analytical solution for the neutral Andronov–Hopf bifurcation curve has been obtained. For the general case of an asymmetric repressilator, neutral curves are found numerically. It is shown that the asymmetric repressor generally is more stable, since the system is oriented to the behavior of the most stable element in the network. Nonlinear dynamic regimes arising in a repressilator with increase of the parameters are studied in detail. It was found that there exists a limit cycle corresponding to relaxation oscillations of protein concentrations. In addition to the limit cycle, we found the slow manifold not associated with above cycle. This is the long-lived transitional regime, which reflects the process of long-term synchronization of pulsations in the work of individual genes. The obtained results are compared with the experimental data known from the literature. The place of the model proposed in the present work among other theoretical models of the repressilator is discussed.

The paper presents a physico-mathematical model of the perturbed region formed in the lower D-layer of the ionosphere under the action of directed radio emission flux from a terrestrial stand of the megahertz frequency range, obtained as a result of comprehensive theoretical studies. The model is based on the consideration of a wide range of kinetic processes taking into account their nonequilibrium and in the two-temperature approximation for describing the transformation of the radio beam energy absorbed by electrons. The initial data on radio emission achieved by the most powerful radio-heating stands are taken in the paper. Their basic characteristics and principles of functioning, and features of the altitude distribution of the absorbed electromagnetic energy of the radio beam are briefly described. The paper presents the decisive role of the D-layer of the ionosphere in the absorption of the energy of the radio beam. On the basis of theoretical analysis, analytical expressions are obtained for the contribution of various inelastic processes to the distribution of the absorbed energy, which makes it possible to correctly describe the contribution of each of the processes considered. The work considers more than 60 components. The change of the component concentration describe about 160 reactions. All the reactions are divided into five groups according to their physical content: ionization-chemical block, excitation block of metastable electronic states, cluster block, excitation block of vibrational states and block of impurities. Blocks are interrelated and can be calculated both jointly and separately. The paper presents the behavior of the parameters of the perturbed region in daytime and nighttime conditions is significantly different at the same radio flux density: under day conditions, the maximum electron concentration and temperature are at an altitude of ~45–55 km; in night ~80 km, with the temperature of heavy particles rapidly increasing, which leads to the occurrence of a gas-dynamic flow. Therefore, a special numerical algorithm are developed to solve two basic problems: kinetic and gas dynamic. Based on the altitude and temporal behavior of concentrations and temperatures, the algorithm makes it possible to determine the ionization and emission of the ionosphere in the visible and infrared spectral range, which makes it possible to evaluate the influence of the perturbed region on radio engineering and optoelectronic devices used in space technology.

Seasonal migrations and herd instinct are traditionally recognized as wild reindeer (Rangifer tarandus L.) species-specific behavioral signs. These animals are forced to overcome water obstacles during the migrations. Behaviour peculiarities are considered as the result of the selection process, which has chosen among the sets of strategies, as the only evolutionarily stable one, determining the reproduction and biological survival of wild reindeer as a species. Natural processes in the Taimyr population wild reindeer are currently occurring against the background of an increase in the influence of negative factors due to the escalation of the industrial development of the Arctic. That is why the need to identify the ethological features of these animals completely arose. This paper presents the results of applying the classical methods of the theory of optimal control and differential games to the wild reindeer study of the migration patterns in overcoming water barriers, including major rivers. Based on these animals’ ethological features and behavior forms, the herd is presented as a controlled dynamic system, which presents also two classes of individuals: the leader and the rest of the herd, for which their models, describing the trajectories of their movement, are constructed. The models are based on hypotheses, which are the mathematical formalization of some animal behavior patterns. This approach made it possible to find the trajectory of the important one using the methods of the optimal control theory, and in constructing the trajectories of other individuals, apply the principle of control with a guide. Approbation of the obtained results, which can be used in the formation of a common “platform” for the adaptive behavior models systematic construction and as a reserve for the cognitive evolution models fundamental development, is numerically carried out using a model example with observational data on the Werchnyaya Taimyra River.