In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space ${\mathbb{R}}^3$, and study the geometric properties of the curve using Caputo’s fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo’s fractional derivative of algebraic polynomials.

In the present work it is indicated that the entropy of the information is the most appropriate tool to analyze the behavior of the flow of prices in the financial market. For this, the following points are mentioned: general concepts of chaos theory applied to the financial market, concept of dynamic systems applied to the flow of prices, time series of prices and the entropy of information applied to the flow of prices in the financial market.

In the present work, necessary and sufficient optimality conditions are presented for the solution to a mínimum time problem. These conditions are established by the principle of the Pontryagin maximum, as a necessary and sufficient condition. In addition, the principle of the Pontryagin maximum is used in the search for the solution to an optimal control problem of mechanical origin. En el presente trabajo se presentan condiciones necesarias y suficientes de optimalidad para la solución a un problema de tiempo mínimo. Estas condiciones son establecidas por el principio del máximo de Pontryagin, como condición necesaria y suficiente. Además, se utiliza el principio del máximo de Pontryagin en la búsqueda de la solución a un problema de control óptimo de origen mecánico.

In this work, we study a numerical method to approximate the exact solution of a simple in situ combustion model. To achieve this, we use the mixed nonlinear complementarity method (MNCP), a variation of the Newton method for solving nonlinear systems, incorporating a single Hadamard product in its formulation. The method is based on an implicit finite difference scheme and a mixed nonlinear complementarity algorithm (FDA-MNCP). One of its main advantages is that it ensures global convergence, unlike the finite difference method and the Newton method, which only guarantee local convergence. We apply this theory to an in situ combustion model, reformulating it in terms of mixed complementarity. Additionally, we compare it with the FDA-NCP method, demonstrating that the FDA-MNCP is computationally more efficient when the spatial discretization is refined.