Home – Home – Jornals – Advanced Search

In this paper, we consider problems of obtaining upper bounds for the components of root-mean-square errors for computational constructions of approximation of an unknown probability density for a given sample. Examples are the computer functional kernel and projection algorithms as well as their important special case - the multidimensional analog of the frequency polygon. These bounds are then used in choosing such versions of kernel and projection algorithms that provide a given level of error in a density approximation.

This study propounded a numerical approach for solving a mathematical model of pollutant spread through forest resources using shifted orthogonal Bernoulli polynomials (OBPs). The model is based on a system of ordinary differential equations, which is transformed into an algebraic system using the collocation approach based on shifted OBPs. Newton's method is employed to obtain numerical solutions, and the results are compared with those obtained using the Runge-Kutta method of fourth-order (RK4) to demonstrate the efficiency of the proposed method. The results demonstrate good agreement with the RK4 method, indicating the proposed method's acceptability for modeling pollutant spread through forest resources.

The article studies a problem of optimal control of heating for a homogeneous half-line. The heating problem is set for a heat conduction equation defined on a half-line where the temperature tends to zero at infinity. At the origin of the spatial coordinate a heat flux, i. e. a non-homogeneous boundary condition of the second kind, is given. The heat flux is modeled using a heating function which is a continuous broken line. This choice of the function is explained by the properties of the technical device under study. The article proves the existence of a solution to such a problem and the uniqueness of its classical solution with a certain error. The optimality of the heating control in this paper means that at the boundary x=0 the temperature at any time is maximum permissible (or, in the first time interval, maximum possible), and at the same time does not exceed a certain critical value, which is chosen to be equal to 1. For the optimal control, a recurrence formula is found at different times, it is proven that this is exactly the optimal solution. That is, at large values of the heat flux the critical temperature at the boundary will be exceeded at some point in time, and at the smaller values the temperature will be lower than that allowed by the material. It is also proven that the heat flux found is the exact upper bound of all admissible heat fluxes for a given discrete control and that such a flux is unique.

The real life problems related to engineering and physical systems are theoretically studied using mathematical models and are generally formulated using linear and non-linear differential equations. This work proposes a numerical technique to find approximate solutions of differential equations utilizing polynomials as base approximation functions and metaheuristic optimization algorithms like Differential Evolution (DE) and Particle Swarm Optimization (PSO) for obtaining the optimal values of coefficients of the polynomials in order to get the desired approximate solution. The algorithms for the proposed method have been executed using MATLAB for computer programming. The effectiveness of the approach suggested in this paper is found to be better than or at least comparable to other numerical methods suggested earlier for solving differential equations.

Numerous higher-order iterative methods have been proposed in the literature for locating roots of nonlinear equations. Among these, methods with optimal order are of particular interest due to their superior efficiency. However, not all of them exhibit consistent performance across all scenarios. Some offer low accuracy, while others suffer from slow convergence, yet there are methods that fail to maintain the desired convergence order in certain applications. This paper aims to address these shortcomings. Consequently, we introduce a novel three-point iterative scheme, whose formulation is based on the widely used two-point King's fourth-order method. This scheme attains eighth-order convergence at the cost of four function evaluations per step. As such, it is optimal according to the Kung-Traub conjecture and boasts an efficiency index of 1.682, which surpasses that of Newton's method and many other higher-order techniques. To assess the methods' performance and validate its theoretical properties, we present several numerical examples. Furthermore, we provide a detailed analysis of the complex dynamics through graphical representations of the basins of convergence, comparing our method with those of other established techniques. The computational results and convergence visualizations confirm that our scheme outperforms existing methods in the literature.

In this article, we are interested in the complex version of Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation for grayscale image inpainting as well as the multi-component Cahn-Hilliard systems for image inpainting, that is an extension approach for color image inpainting. We have studied well-posedness of the steady state problem associated to the complex Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation as well as to Bertozzi-Esedoglu-Gillete-Cahn Hilliard systems. Then, Backward Euler discretization on time has been considered in both models mentioned above. We were able to prove the stability of the backward Euler scheme. Finally, we do some numerical simulations that confirm the theoretical results and show the efficiency of the scheme. The simulations were done using FreeFem++.

A method is proposed for restoring two coefficients in a mixed boundary value problem for an inhomogeneous hyperbolic partial differential equation of the second order based on additional information about the solution of the boundary value problem for a given fixed value of the spatial argument of the solution. The problem simulates the propagation of small transverse vibrations of a finite string with an end affected by the gravity of a body with varying mass. The proposed algorithm provides a sequential restoration of the multiplier in the heterogeneity of the oscillation equation and the coefficient in the nonclassical boundary condition based on the values of one additional function of one argument.

The article considers the possibility of applying Karl Popper’s principle of falsification to mathematics. A common position is that the principle of falsification can only be applied to the empirical sciences, since statements not about the surrounding world are inherently unfalsifiable. The purpose of this article is to demonstrate that applying the principle of falsification to mathematics can be pragmatically interesting and feasible. The first part of the article discusses the principle of falsification itself and why the question of the falsifiability of mathematics is basically worthy of study. The second part presents an argument that the main cause for the apparent unfalsifiability and the lack of scientific status of mathematics is the implicit acceptance of traditional mathematical Platonism by researchers. The third part examines the possibility of employing abstrаction principles to transform traditional Platonism into a version of naturalism. By adopting the metaphysics and epistemology of such position, mathematics can be considered falsifiable on par with other sciences.

Throughout the human history, innumerable religious, philosophical, and scientific models of our world have been developed. But, to date, no model of the Universe has been proposed that would be supported by all the confessions of modern society. However, recently, the advent of such a tool as the computer has enabled the rapid development of artificial intelligence. In this paper, the new model of the Universe is based on advances in computer technology, information processing, and AI development, as well as new astrophysical data, and also previously established philosophical and theological approaches were involved.

The article considers a technotropic approach to the development of artificial intelligence (AI) that is focused on creating systems capable of conscious activity. In the context of the dominance of traditional methods and optimization research, the author proposes a transdisciplinary paradigm comprising four levels: philosophical (an axiomatic system), methodological (integration of principles into the phenomenon of jimination generalizing and expanding the recursion mechanism), formal (conceptualization and formalization of jimination with an example of application to Gоdel’s theorem), and applied (comparison of jimination with some modern AI technologies). The concept of technotropy is analogous to the concept of anthropy and emphasizes the possibility of realizing the phenomenon of self-organization by technology with the correct approach. This work lays the theoretical and practical foundation for the development of a new generation of AI that goes beyond existing models and offers a methodological basis for further research into conscious systems in artificial intelligence.