After introducing some basic mathematical theory on kleinian groups, we discuss both technical features and drawbacks of the classic approach adopted to render the digital representation of diagrams that display either the action of tessellation and the limit sets related to Kleinian groups in one complex variable. We will then follow to illustrating the details of a new approach, based upon numerical base conversion. This discussion will be supported by pseudo-code, in order to explain the implementation guidelines. Finally, we will review the benefits of this algorithm.
The work is devoted to the construction and analysis of the gradient method based on a modified explicit second-order Runge--Kutta method, constructed using the Lagrange--Burmann expansion. A two-step method with inertia based on the heavy ball method is proposed. Convergence theorems are proven for strongly convex quadratic and perturbed quadratic functions. Analytical expressions for the optimal parameters of the method are obtained. For the quadratic function it is demonstrated, that the proposed method converges faster, than other well-known accelerated methods. The results of the application of the method to the numerical solution of linear and nonlinear boundary value problems (Dirichlet problem for a 3D Poisson equation, problems from the calculus of variations, problems for integro-differential equations) are presented. It is demonstrated that, in comparison with well-known methods, the proposed method allows one to obtain a numerical solution with the required accuracy for different grid resolutions in a smaller number of iterations and time.
In various fields, artificial intelligence (AI) models are increasingly used for decision-making based on machine learning. In metric machine learning methods, objects are treated as precedents, and only one operation is used: determining the similarity (difference) between these precedents and an unknown object. The main limitation of existing metric methods is related to representing a common feature space for all objects and, consequently, a single measure for measuring distances between objects. This limitation is overcome by constructing a unique local feature space for each object and finding individual measures that determine the hierarchy of its similarity to other objects, relevant to the given context. The article discusses the problem of analyzing sets of objects with local descriptions and proposes a solution using d(S)-metrics, which reflect differences in distance series between the same objects but in different local spaces. Introducing d(S)-metrics allows for further visual analysis of compositions of objects with local descriptions using multidimensional metric scaling. The article provides a practical example of such analysis in the task of recognizing vehicle types based on geometric features of their silhouettes.
The problem of synthesizing an adaptive robust controller using a tensor representation of a quasi-linear parameter-varying model is considered. This type of approximation of nonlinear dynamical systems can be reduced to convex polytopic forms in over a given parameter range. As a result, convex programming methods are applicable to them, including formulation of the problem in terms of linear matrix inequalities. The theory of H-infinity-optimal control is used as a technique for the synthesis of robust controllers. The result of the synthesis is a tensor model that describes a family of robust controllers that ensure stability and controllability of the object for the entire range of varied parameters.
The article reveals the effect of the influence of pairwise non-intersecting and not passing through the origin of coordinates of non-isolated singularities in the lowest coefficient of the generalized Cauchy-Riemann equation on the formulation of boundary value problems. The condition at the boundary of the region turned out to be insufficient to solve such problems. Therefore, we considered a case that combines elements of the Riemann-Hilbert problems on the boundary of a domain and linear conjugation on the surrounding area.
