An optimization of the traversal algorithm using tetrahexacontree data structure is presented, which improves raytracing performance in voxel scenes. In existing research on this data structure, the use of the classical 3DDDA method [1] was proposed for traversing the cells of the space partition without any additional modifications. This approach proved to be less efficient in terms of performance compared to the popular hierarchical voxel data structure—the octree [2]. The aim of this work is to justify and evaluate the application of optimization techniques to the tetrahexacontree traversal algorithm. By analyzing cell occupancy bitmaps, the ray can skip empty regions within the tree nodes, thereby reducing the number of ray steps and thus improving performance. The optimization incurs a small overhead in terms of processor time and requires a fixed amount of extra memory, but does not require any additional preprocessing of the input data. The optimization is compatible with both dense and sparse tree representations. On average, the algorithm optimization yields a 28
Heat transfer in a thermodynamically nonequilibrium anisotropic half-space with transverse anisotropy is modeled using a generalized wave heat transfer law. The law retains the mathematical structure of the classical Fourier heat conduction equation, but incorporates a time argument delay equal to the relaxation time. The resulting governing equation is of parabolic type with a time delay corresponding to the relaxation time. For the three-dimensional unsteady heat conduction problem with tensorial transport coefficients, a novel analytical solution is derived by applying the double Fourier transform with respect to two spatial coordinates and the Laplace transform with respect to time. The computational results reveal that the isothermal surfaces are approximated by elliptic paraboloids with local inflection lines, while the temporal temperature profiles exhibit propagating fronts of jump discontinuities, thereby confirming the wave-like nature of heat transfer.
An optimal control problem for a high-order linear pseudohyperbolic equation with a right-hand side control dependent only on time is considered. The equation is reduced to a system of two partial differential equations. The needle variation method is used to obtain necessary optimality conditions in the form of Pontryagin’s maximum principle. The uniqueness of a weak solution is established.
This paper presents the construction and investigation of a mathematical model of an aerodynamic pendulum in a medium flow. To model the aerodynamic forces, the hypothesis of quasi-static flow around a plate by the medium is employed. The equations of motion are obtained in dimensionless variables, describing the regimes of small oscillations of the aerodynamic pendulum. A violation of uniqueness in determining the angle of attack is demonstrated. A graphical and numerical investigation of the solutions to the equilibrium equations is carried out, and various nontrivial stationary points are found. For the identified stationary points, stability regions are constructed using the Routh–Hurwitz method. A software package has been developed in the MATLAB mathematical environment, enabling the construction of stability regions and the numerical integration of the equations describing the body’s oscillations.
The paper examines modern approaches to efficient storage and analysis of time series arising in monitoring industrial equipment, in particular, CNC machines. The main problem is the growth of data amount, which requires both the optimization of storage systems and the reduction of the dimension of input features without loss of information content. To solve the problem, we conducted a comparative analysis of purpose-built database management systems (DBMS): ClickHouse, InfluxDB, TimescaleDB, and PostgreSQL. Based on the testing results for ingestion rate, scalability, and resource consumption, ClickHouse was chosen. In parallel, we studied feature selection methods based on machine learning: L1 regularization, random forest, and gradient boosting. By using a real dataset of 18 material (wax) milling experiments, it was shown that gradient boosting (LightGBM) provides the best accuracy and robustness in feature importance evaluation. As a result, 20 most significant features related to position, speed, current, and voltage along the X, Y, and spindle S axes were identified. Based on these features, the condition of the tool, the completion of machining, and visual inspection results can be predicted with high reliability. The proposed approach allows one to reduce the amount of stored data, improve query performance, and reduce hardware costs for storing and processing time series.
We propose a new approach to optimization problems with uncertain factors, which is related to transformation of the initial problem into a bi-criteria optimization problem. This problem is solved by combining the successive concessions method and linear convolution of the criteria. An application of the method to the simplest extremal graph problems is described.
An abstract algebraic approach to the formation of high-dimensional matrix games is proposed. The considered class of games includes, in particular, statistical games. A general method for constructing randomized decision rules for the problem of estimating a parameter of a discrete random variable is discussed. For economically significant tasks, the cost of an additional observation is calculated.
The paper considers a differential game model with functionals represented by a convolution in the form of the minimum of two criteria, one of which describes the competition of players in a common (external) sphere of activity and the other describes the personal achievements of each player (in the internal sphere). The player’s control is the resource redistribution between the external and internal spheres. It is shown that, under some natural assumptions of monotonicity of criteria in such games, Nash and Stackelberg equilibria exist and coincide, possessing the properties of stability and Pareto optimality.
A dynamic game related to a resource management problem with asymmetric players is considered. The resource evolution depends on interchanged usage regimes: exploitation periods, where many players extract a common resource, and moratorium periods, where extraction is prohibited and the resource stock evolves according to the natural growth rule. Both noncooperative and cooperative behaviors of players are constructed and compared. To maintain sustainable resource exploitation, a certain ratio of the lengths of moratorium to extraction periods is applied. The value of exploitation horizon is determined by optimizing the price of anarchy.
A cooperative differential game on a hypergraph is considered. A characteristic function of special type is introduced taking into account the network structure of the game, and its properties are studied. The Shapley value is used as a cooperative optimality principle. For games on hypergraphs with a cycle-free line graph, an explicit formula for the Shapley value is obtained. An illustrative example is considered.
The paper considers one class of finite non-cooperative games (with a finite number of strategies for each player)—E.B. Yanovskaya’s polymatrix games. More specifically, three-player polymatrix games, so-called hexamatrix games (HMGs), which can be completely described by six matrices, are studied. A number of model examples of three-party conflicts, describing some real-life situations, are presented and formulated as HMGs. The feasibility of using hexamatrix games to model economic relationships between three participants is demonstrated. To find the Nash equilibrium in the formulated games, an optimization approach is used, where the equilibrium problem is reduced to a nonconvex optimization problem with a bilinear structure. The latter is solved using A.S. Strekalovskii’s Global Search Theory (GST) for (d.c.) optimization problems with objective functions representable as the difference of two convex functions.
This paper considers the congestion game with flow constraints. While the total number of players in the congestion game is usually specified and the flow of players assigned to each of the alternatives is, generally speaking, unlimited, in the formulation considered in this paper, the flow of players can be limited for each of the available alternatives and in total. The paper provides a general formulation of the congestion game with flow constraints and studies its solution space. We estimate the price of anarchy for different numbers of players, which helps us determine when the game’s equilibrium assignment is close to the social optimum and when it deviates. Finally, we consider examples of practical problems and cases that can be modeled and described using the corresponding game.
This paper examines games on networks with linear best responses, which allow for the analysis of how interaction structures influence agents’ strategic behavior. Special attention is given to intervention issues in such models, particularly in selecting optimal intervention strategies aimed at maximizing the central planner’s objective function. Two main control policies are analyzed: individual agent incentives and modifications of the interaction structure. The concept of a representative agent is introduced to simplify equilibrium analysis and control problems in games on networks. Both aggregate outcome maximization problems and adversarial scenarios between competing central planners are considered. Analytical conditions are derived to determine whether controlling the interaction structure is more effective than influencing individual incentives. Numerical experiments confirm the theoretical results and demonstrate their applicability to different types of network structures.
The paper proposes a modified concealed voter model MCVM in which each agent in the network is represented by two nodes corresponding to her external (public) and internal (hidden) opinion. Agents exchange opinions in the external layer and copy the internal opinion into the external layer and vice versa. In our work, one more action has been added—transmitting the external opinion of an agent to another agent in the internal layer. We find a formula to calculate the average time to reach consensus and compare them for MCVM and CVM. The results of numerical simulations of opinion dynamics according to the proposed MCVM model are presented.
In the article, the European option superhedging problem is identified with a dynamic stochastic zero-sum game between the market and the contract seller. The seller manages a portfolio of underlying assets in order to minimize her expected exponential risk. The market determines a probability distribution for discounted prices of traded assets: absolutely continuous with respect to a given underlying distribution and maximazing seller’s expected risk. Recurrence relations for the upper and lower values of the game are obtained. It is shown that the absence of arbitrage opportunities in the market is a necessary and sufficient condition for the existence of a self-financing portfolio, with which the lower bound in the definition of the upper value of the game is obtained. Such a portfolio is superhedging, and the upper value of the game allows one to calculate the upper hedging price. Moreover, it is shown that, in a market model without arbitrage opportunities, there is always a game equilibrium. The saddle point of the game, if it exists, determines a superhedging portfolio and a martingale probability distribution, with which an upper bound in the definition of the upper value of the game is achieved. This distribution defines the seller’s worst market in the sense that the reserve of the superhedging portfolio is fully consumed in that market model. Using examples, we provide comparison between results of option calculations based on probabilistic and trajectory-based game approaches, analyze advantages and disadvantages of the topology choice ( σ (L^1,L^∞) topology instead of weak topology) in the probabilistic formulation of this type of problems.
In this paper, we consider two levels of cooperation in a differential game with pairwise interactions. At the first level, the partner sets are treated as players, and the Shapley value is used to allocate the payoff among the partner sets. At the second level, this payoff is distributed among the individual players within each partner set. A new characteristic function is constructed, and its convexity is proved. The results are illustrated through a pollution control problem involving pairwise interactions.
The paper describes an algorithm for finding the Wardrop equilibrium and optimal distributions of traffic flows in an urban road network. A software code has been developed to implement this algorithm. A procedure is described for conducting numerical experiments and analyzing their results to study the computational complexity of the algorithm depending on the model dimension.
A model of opinion dynamics is considered, in which the trust between the agents is unknown and modeled using random variables with certain probability distributions. Additionally, there is a player whose goal is to maintain the agents’ opinions at a specific level. Initially, an optimal control is found in explicit form, assuming that the trust coefficients are known. Then this control is used at each step to obtain realizations of the random variables. Computer experiments have been conducted.
Variational inequalities (VIs) serve as a powerful tool for various problems. This setting can be applied to a wide range of optimization, machine learning (ML), and other challenges. At the same time, large volumes of data are essential for high performance in ML tasks, which are addressed through stochastic approaches. However, widely used SGD method suffers from a non-decreasing variance of the stochastic gradient. To resolve this issue, variance reduction techniques were developed. This approach is well-studied for minimization but less extensively for VIs. In this paper, we modify the SAGA method, known for its effectiveness in stochastic minimization, by integrating Extragradient to address VI problems. We provide a theoretical analysis of the proposed method and conduct experiments, including bilinear problems and image denoising tasks.
While deep encoder-decoder models dominate endoscopic segmentation, their reliance on full fine-tuning or training from scratch is computationally expensive and data-intensive. This paper challenges this by demonstrating that an extremely efficient model—a frozen foundation model encoder with a shallow decoder—can achieve state-of-the-art performance. Our core contribution is a systematic, layer-wise analysis to identify the single most effective feature source within the encoder’s hierarchy, challenging the common practice of using the final, most abstract layer. We identify a distinct performance peak at an intermediate layer (Layer 12), with an optimal trade-off between high-level semantic understanding and the high-resolution spatial fidelity crucial for segmentation. Despite training only 650k parameters, our method surpasses existing benchmarks on the challenging multi-center PolypGen dataset with a Dice score of 0.972. This work provides an evidence-based methodology for efficient feature extraction, significantly lowering the computational and data barriers for developing high-performance clinical AI tools.
