In this paper, we apply the method of geometrization of random vectors [1] to turbulent media, which we understand as random vector fields on base manifolds. This gives rise to various geometric structures on the tangent as well as cotangent bundles. Among these, the most important is the Mahalanobis metric on the tangent bundle, which allows us to obtain all the necessary ingredients for implementing the scheme [2] to the description of flows in turbulent media. As an illustration, we consider the applications to flows of real gases based on Maxwell–Boltzmann statistics.

We find the twisted extensions of the symmetry algebra of the 2D Euler equation in the vorticity form and use them to construct new Lax representation for this equation. Then we generalize this result by considering the transformation Lie--Rinehart algebras generated by finite-dimensional subalgebras of the symmetry algebra and derive a family of Lax representations for the Euler equation. The family depends on functional parameters and contains a non-removable spectral parameter.

The problem of identifying the input-output mapping of multidimensional discrete-time stochastic systems using an information-theoretic criterion is considered. As an information-theoretic measure of dependence, which is the basis for constructing the corresponding criterion for statistical linearization, a measure based on the symmetric Kullback-Leibler divergence is proposed, in contrast to the usual Kullback-Leibler divergence, which, as is well known, is not symmetric. Based on such symmetric divergence, a corresponding measure of stochastic dependence is constructed, and the criterion of statistical linearization is the condition of pairwise coincidence of this measure of dependence, calculated for each component of the vector of the output variable of the system and each component of the vector of the input variable of the system, on the one hand, and the corresponding components of the output vector variable of the linearized model and the vector of the input variable of the system. This approach allows us to construct explicit analytical expressions for the weighting coefficients of the linearized model, which, at the same time, has the property of complete equivalence to the nonlinear system under study in the sense of coincidence of the corresponding information-theoretic characteristics.

Abstract: With the increasing adoption of machine learning systems, concerns around bias and privacy have gained significant research interest. This work investigates the intersection of algorithmic fairness and differential privacy by evaluating differentially private fair representations. The LAFTR framework aims to learn fair data representations while maintaining utility. Differential privacy is injected into model training using DP-SGD to provide formal privacy guarantees. Experiments are conducted on the Adult, German Credit, and CelebA datasets, with gender and age as sensitive attributes. The models are evaluated across various configurations, including the privacy budget epsilon, adversary strength, and dataset characteristics. Results demonstrate that with proper tuning, differentially private models can achieve fair representations comparable or better than non-private models. However, introducing privacy reduces stability during training. Overall, the analysis provides insights into the tradeoffs between accuracy, fairness and privacy for different model configurations across datasets. The results establish a benchmark for further research into differentially...

The paper presents the methods for intelligent forecasting of the state of industrial processes, for scenario modeling of production situations, and for the intelligent synthesis of proactive management systems. Associative models based of inductive knowledge are used.