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Theory of anomalous diffusion, which describe a vast number of transport processes with power law mean squared displacement, is actively advancing in recent years. Diffusion of liquids in porous media, carrier transport in amorphous semiconductors and molecular transport in viscous environments are widely known examples of anomalous deceleration of transport processes compared to the standard model.

Direct Monte Carlo simulation is a convenient tool for studying such processes. An efficient stochastic simulation algorithm is developed in the present paper. It is based on simple renewal process with interarrival times that have power law asymptotics. Analytical derivations show a deep connection between this class of random process and equations with fractional derivatives. The algorithm is further generalized by coupling it with chemical reaction simulation. It makes stochastic approach especially useful, because the exact form of integrodifferential evolution equations for reaction — subdiffusion systems is still a matter of debates.

Proposed algorithm relies on non-markovian random processes, hence one should carefully account for qualitatively new effects. The main question is how molecules leave the system during chemical reactions. An exact scheme which tracks all possible molecule combinations for every reaction channel is computationally infeasible because of the huge number of such combinations. It necessitates application of some simple heuristic procedures. Choosing one of these heuristics greatly affects obtained results, as illustrated by a series of numerical experiments.

An approximate mathematical model of blood flow in an axisymmetric blood vessel is studied. Such a vessel is understood as an infinitely long circular cylinder, the walls of which consist of elastic rings. Blood is considered as an incompressible fluid flowing in this cylinder. Increased pressure causes radially symmetrical stretching of the elastic rings. Following J. Lamb, the rings are located close to each other so that liquid does not flow between them. To mentally realize this, it is enough to assume that the rings are covered with an impenetrable film that does not have elastic properties. Only rings have elasticity. The considered model of blood flow in a blood vessel consists of three equations: the continuity equation, the law of conservation of momentum and the equation of state. An approximate procedure for reducing the equations under consideration to the Korteweg – de Vries (KdV) equation is considered, which was not fully considered by J. Lamb, only to establish the dependence of the coefficients of the KdV equation on the physical parameters of the considered model of incompressible fluid flow in an axisymmetric vessel. From the KdV equation, by a standard transition to traveling waves, ODEs of the third, second and first orders are obtained, respectively. Depending on the different cases of arrangement of the three stationary solutions of the first-order ODE, a cnoidal wave and a soliton are standardly obtained. The main attention is paid to an unbounded periodic solution, which we call a degenerate cnoidal wave. Mathematically, cnoidal waves are described by elliptic integrals with parameters defining amplitudes and periods. Soliton and degenerate cnoidal wave are described by elementary functions. The hemodynamic meaning of these types of decisions is indicated. Due to the fact that the sets of solutions to first-, second- and third-order ODEs do not coincide, it has been established that the Cauchy problem for second- and third-order ODEs can be specified at all points, and for first-order ODEs only at points of growth or decrease. The Cauchy problem for a first-order ODE cannot be specified at extremum points due to the violation of the Lipschitz condition. The degeneration of the cnoidal wave into a degenerate cnoidal wave, which can lead to rupture of the vessel walls, is numerically illustrated. The table below describes two modes of approach of a cnoidal wave to a degenerate cnoidal wave.

The SAS program is the primary data processing tool for the YuMO small-angle neutron scattering spectrometer. The paper presents a retrospective analysis of its two-decade evolution, from a Fortran prototype to a modern software system. The analysis focuses on the architectural decisions that have ensured the program’s long-term viability and its ability to adapt to instrument upgrades.

The core solution was a modular architecture that abstracts the detector system. This enabled the seamless integration of data from two scattering detectors and, later, from a position-sensitive detector. A strict processing pipeline and a unified internal data representation formed the basis for physically grounded algorithms, including weighted merging of spectra, resolution-aware smoothing, and built-in statistical quality control. The program’s interfaces—a command line for batch processing and a graphical user interface for interactive work—are built upon a single computational core, ensuring result consistency and flexibility in use.

Long-term operation has confirmed that the underlying architectural principles naturally align with the key characteristics of international software quality standards, particularly those critical for long-term sustainability. Therefore, the development and evolution of SAS demonstrates a universal set of architectural principles that can serve as a foundation for building sustainable scientific software in related fields of experimental physics.

Methods for establishing standards of environmental quality by data of ecological monitoring are proposed. These are: methods of bioindication by indices of species diversity and size structure of communities, by indices of fish productivity; method for searching for reasons of environmental trouble and ranking them by their contribution into the trouble; methods for standardization of factors which are important as causes of environmental trouble.

Since 1950s the field of city transport modelling has progressed rapidly. The first equilibrium distribution models of traffic flow appeared. The most popular model (which is still being widely used) was the Beckmann model, based on the two Wardrop principles. The core of the model could be briefly described as the search for the Nash equilibrium in a population demand game, in which losses of agents (drivers) are calculated based on the chosen path and demands of this path with correspondences being fixed. The demands (costs) of a path are calculated as the sum of the demands of different path segments (graph edges), that are included in the path. The costs of an edge (edge travel time) are determined by the amount of traffic on this edge (more traffic means larger travel time). The flow on a graph edge is determined by the sum of flows over all paths passing through the given edge. Thus, the cost of traveling along a path is determined not only by the choice of the path, but also by the paths other drivers have chosen. Thus, it is a standard game theory task. The way cost functions are constructed allows us to narrow the search for equilibrium to solving an optimization problem (game is potential in this case). If the cost functions are monotone and non-decreasing, the optimization problem is convex. Actually, different assumptions about the cost functions form different models. The most popular model is based on the BPR cost function. Such functions are massively used in calculations of real cities. However, in the beginning of the XXI century, Yu. E. Nesterov and A. de Palma showed that Beckmann-type models have serious weak points. Those could be fixed using the stable dynamics model, as it was called by the authors. The search for equilibrium here could be also reduced to an optimization problem, moreover, the problem of linear programming. In 2013, A.V.Gasnikov discovered that the stable dynamics model can be obtained by a passage to the limit in the Beckmann model. However, it was made only for several practically important, but still special cases. Generally, the question if this passage to the limit is possible remains open. In this paper, we provide the justification of the possibility of the above-mentioned passage to the limit in the general case, when the cost function for traveling along the edge as a function of the flow along the edge degenerates into a function equal to fixed costs until the capacity is reached and it is equal to plus infinity when the capacity is exceeded.

The work presents an experimental study of the tree structure that occurs during a medical examination. At each meeting with a medical specialist, the patient receives a certain number of areas for consulting other specialists or for tests. A tree of directions arises, each branch of which the patient should pass. Depending on the branching of the tree, it can be as final — and in this case the examination can be completed — and endless when the patient’s goal cannot be achieved. In the work both experimentally and theoretically studied the critical properties of the transition of the system from the forest of the final trees to the forest endless, depending on the probabilistic characteristics of the tree.

For the description, a model is proposed in which a discrete function of the probability of the number of branches on the node repeats the dynamics of a continuous gaussian distribution. The characteristics of the distribution of the Gauss (mathematical expectation of $x_0$, the average quadratic deviation of $\sigma$) are model parameters. In the selected setting, the task refers to the problems of branching random processes (BRP) in the heterogeneous model of Galton – Watson.

Experimental study is carried out by numerical modeling on the final grilles. A phase diagram was built, the boundaries of areas of various phases are determined. A comparison was made with the phase diagram obtained from theoretical criteria for macrosystems, and an adequate correspondence was established. It is shown that on the final grilles the transition is blurry.

The description of the blurry phase transition was carried out using two approaches. In the first, standard approach, the transition is described using the so-called inclusion function, which makes the meaning of the share of one of the phases in the general set. It was established that such an approach in this system is ineffective, since the found position of the conditional boundary of the blurred transition is determined only by the size of the chosen experimental lattice and does not bear objective meaning.

The second, original approach is proposed, based on the introduction of an parameter of order equal to the reverse average tree height, and the analysis of its behavior. It was established that the dynamics of such an order parameter in the $\sigma = \text{const}$ section with very small differences has the type of distribution of Fermi – Dirac ($\sigma$ performs the same function as the temperature for the distribution of Fermi – Dirac, $x_0$ — energy function). An empirical expression has been selected for the order parameter, an analogue of the chemical potential is introduced and calculated, which makes sense of the characteristic scale of the order parameter — that is, the values of $x_0$, in which the order can be considered a disorder. This criterion is the basis for determining the boundary of the conditional transition in this approach. It was established that this boundary corresponds to the average height of a tree equal to two generations. Based on the found properties, recommendations for medical institutions are proposed to control the provision of limb of the path of patients.

The model discussed and its description using conditionally-infinite trees have applications to many hierarchical systems. These systems include: internet routing networks, bureaucratic networks, trade and logistics networks, citation networks, game strategies, population dynamics problems, and others.

Approbation of calculation of borders of water quality classes for the purpose of ecological diagnosis and standardization by data of the Rybinsk reservoir is carried out. For bioindication indicators of phytoplankton fluorescence and the contents of pigments of phytoplankton are used. Chesnokov's importance coefficient proved to be the most preferred measure of connection for analyzing the effects of environmental factors on indicators. The factors important for environmental condition are identified. Comparison of borders between quality classes “valid” and “invalid” of factors values and boundaries of the classifications of water quality.

The article considers the problem of the stability of the vertical position of a Maxwell pendulum during its periodic up-down movements. Two types of transition movements are considered: “stop” — occurs when the body of the pendulum in its highest position on the string (during its “standard” upward movement) stops for a moment; “two-link pendulum” — occurs when the entire thread from the body of the pendulum is selected (the lowest position of the body on the thread during its “standard” downward movement), and the body is forced to rotate relative to the thread around the point of its attachment to the body. It is shown that for any values of the pendulum parameters, this position is unstable in the sense that oscillations of the thread around the vertical of finite amplitude occur in the system for arbitrarily small initial deviations. In addition, it has been established that no shock phenomena occur during the movement of the Maxwell pendulum, and the model of this pendulum itself, with the values of its parameters often used in the literature, is incorrect according to Hadamard. In this work, it is shown that the vertical position of the pendulum threads during the indicated oscillatory movements of the body along the threads for any non-degenerate values of the parameters of the Maxwell pendulum is always unstable in the above sense. Moreover, this instability is caused precisely by transitional movements of the 2nd type. In this work, it is further shown that no jumps in speeds or accelerations (due to which shocks or “jerks” in the tension of the threads can occur) do not occur during the indicated movements of the Maxwell pendulum model under consideration. In our opinion, the “jerks” observed in the experiments are due to other reasons, for example, the technical imperfection of the instruments on which the experiments were carried out.

In single-photon emission computed tomography (SPECT), patients with bone disorders receive a radiopharmaceutical (RP) that accumulates selectively in pathological lesions. Accurate quantification of RP uptake plays a critical role in disease staging, prognosis, and the development of personalized treatment strategies. Traditionally, the accuracy of quantitative assessment is evaluated through in vitro clinical trials using the standardized physical NEMA IEC phantom, which contains six spheres simulating lesions of various sizes. However, such experiments are limited by high costs and radiation exposure to researchers. This study proposes an alternative in silico approach based on numerical simulation using a digital twin of the NEMA IEC phantom. The computational framework allows for extensive testing under varying conditions without physical constraints. Analogous to clinical protocols, we calculated the recovery coefficient (RC_max), defined as the ratio of the maximum activity in a lesion to its known true value. The simulation settings were tailored to clinical SPECT/CT protocols involving ^99mTc for patients with bone-related diseases. For the first time, we systematically analyzed the impact of lesion-to-background ratios and post-reconstruction filtering on RC_max values. Numerical experiments revealed the presence of edge artifacts in reconstructed lesion images, consistent with those observed in both real NEMA IEC phantom studies and patient scans. These artifacts introduce instability into the iterative reconstruction process and lead to errors in activity quantification. Our results demonstrate that post-filtering helps suppress edge artifacts and stabilizes the solution. However, it also significantly underestimates activity in small lesions. To address this issue, we introduce post-reconstruction correction factors derived from our simulations to improve the accuracy of quantification in lesions smaller than 20 mm in diameter.

Dam safety assessment increasingly relies on continuous monitoring of hydrometeorological variables; however, identifying early-stage instability remains challenging due to complex spatio-temporal interactions and highly imbalanced failure observations. This study proposes a deep learning framework based on a Convolutional Bidirectional Gated Recurrent Unit (CNN–BiGRU) architecture to learn spatio-temporal precursors of dam instability from multivariate hydrometeorological time series. The convolutional component extracts localized temporal patterns associated with short-term fluctuations, while the bidirectional recurrent structure captures long-range dependencies and evolving dynamics preceding critical states.

The proposed model is evaluated on a real-world dam monitoring dataset comprising multiple water-level, meteorological, and derived dynamic indicators. To address class imbalance, a cost-sensitive training strategy using class weighting is adopted without synthetic oversampling. Experimental results demonstrate strong predictive performance, achieving an accuracy of 0.961, precision of 0.901, recall of 0.757, and an F1-score of 0.823. The model further attains a ROC-AUC of 0.907 and a PR-AUC of 0.819, indicating robust discrimination capability under imbalanced conditions.

Feature importance analysis reveals that short- and medium-term water level variability, including rolling standard deviation, volatility, and multi-scale gradients, play a dominant role in characterizing pre-instability behavior, providing physically interpretable insights into dam response dynamics. The findings suggest that the CNN–BiGRU framework effectively captures meaningful spatio-temporal precursors and offers a reliable data-driven tool for supporting dam safety monitoring and decision-making under real operational conditions.