http://hdl.handle.net/10366/4137 2026-04-25T22:39:03Z 2026-04-25T22:39:03Z Ferragut Canals, Luis http://hdl.handle.net/10366/169391 2026-01-31T01:00:37Z 2026-01-01T00:00:00Z

[ES]Este libro recoge los apuntes actualizados de las enseñanzas que formaban parte de la asignatura Ampliación de Matemáticas que impartí en la Escuela Técnica Superior de Ingenieros de Minas de Madrid en los años 80. El objetivo de estos apuntes era proporcionar a los alumnos las definiciones precisas de los conceptos y las propiedades de éstos para poder abordar el cálculo vectorial y tensorial. Una vez realizado este trabajo la realización de cálculos es un trabajo mecánico. Con ello se pretende que se comprendan bien expresiones en los que intervienen la diferenciales y se realicen los cálculos correctamente con conocimiento de su significado. Consideremos un ejemplo: En la mayoría de los libros de física e ingeniería es habitual llamar diferencial de área a una pequeña parte de una superficie y se razona despues con este concepto ambiguo. Lo que aportan las matemáticas es una definición precisa de este concepto introduciendo el diferencial de área como una forma de orden 2 de modo que al aplicarla a dos vectores nos da el área del paralelógramo formado por éstos. A su vez las formas de orden 2 se construyen a partir de formas de orden 1 (elementos de un espacio dual) mediante el producto exterior, operación bien definida. El manejo de conceptos y operaciones precisas y bien definidas permite razonar y hacer cálculos sin peligro de cometer errores. El curso presupone que se tienen los conocimientos básicos de un primer curso de cálculo infinitesimal de una variable real y de álgebra lineal así como algunos conceptos de topología. El cálculo vectorial es álgebra en el espacio tangente en cada punto de un dominio para luego integrar los resultados en cada punto (en definitiva sumar) cuando recorremos todos los puntos del dominio: La primera parte de este libro se dedica al cálculo diferencial (capítulo 1) y a la integración (capítulo 2) en varias dimensiones donde se generalizan los resultados del cálculo en una variable real. El capítulo 3 está dedicado álgebra tensorial donde se estudia el concepto de tensor y sus aplicaciones geométricas. En el capítulo 4 se construye el espacio tangente introduciendo la noción de vector tangente como una derivación (como se hace en geometría diferencial), vemos ejemplos de vectores tangentes y aprendemos a realizar cambios de coordenadas. En una segunda sección se introducen las formas diferenciales a partir del espacio dual del espacio tangente. En el capítulo 5 construimos los tensores diferenciales y a partir del espacio tangente y el espacio de tensores diferenciales se introduce la noción de campo vectorial y campos de formas. En particular se estudia la operación de diferenciación exterior de formas. En el capítulo 6 se introduce la noción de cadena y la integración de formas en cadenas. El capítulo 7 está dedicado al teorema general de Stokes en cadenas y sus aplicaciones y se particulariza a los tres teorema clásicos de Stokes, teorema de Green, teorema de la divergencia y el teorema de Gauss-Ostrogradski que son aquí una consecuencia del teorema general. Finalmente en el capítulo 8 se deducen algunas ecuaciones de la mecánica y más en particular de la mecánica de medios continuos. Más precisamente, obtendremos la variación de los distintos objetos geométricos, es decir funciones, campos, formas y en general tensores por acción de un campo vectorial representando la velocidad de un fluido asociado a un grupo uniparamétrico de transformaciones. La herramienta básica es el concepto de derivada de Lie de la geometría diferencial que introducimos limitándonos aquí a regiones del espacio euclídeo. He completado cada capítulo de estos apuntes con ejercicios cuya solución se da al final del libro con la esperanza de que los estudiantes intenten resolverlos por su cuenta antes de mirar la solución.

2026-01-01T00:00:00Z Jaiswal, Alshwarya Sunil, Kumar Ramos Calle, Higinio http://hdl.handle.net/10366/166848 2025-08-30T00:11:45Z 2025-01-01T00:00:00Z

[EN] This article is concerned with the construction and analysis of efficient uniformly convergent methods for a class of parabolic systems of coupled singularly perturbed reaction–diffusion problems with discontinuous source term. Due to the discontinuity in the source term, the solution to this problem exhibits interior layers along with boundary layers, which are overlapping and interacting in nature. To achieve an efficient numerical solution for the coupled system under consideration, at interior points (excluding the interface point) we employ a special finite difference scheme in time (where the components of the approximate solution are decoupled at each time level) and the central difference scheme in space; for mesh points on the interface, a special finite difference scheme decoupling the components of the approximate solution is developed. A rigorous error analysis is provided, establishing the method’s uniform convergence. In terms of computational cost, our numerical methods are more efficient than existing approaches for solving this class of problems. Finally, we provide numerical results to substantiate the theory and showcase the efficiency of our methods.

2025-01-01T00:00:00Z Moaaz, Osama Ramos Calle, Higinio http://hdl.handle.net/10366/166842 2025-08-30T00:11:21Z 2024-01-01T00:00:00Z

[EN] In this work, only two independent conditions for the oscillation of all solutions of evenorder delay differential equations in the non-canonical case are established. Using comparison techniques with first- and second-order delay differential equations, we obtain easyto- apply criteria that improve previous results in the literature. In addition, we show the importance of our results by applying them to examples that have been frequently used in related works.

2024-01-01T00:00:00Z Kumar, Shashikant Kumar, Sunil Ramos Calle, Higinio Kuldeep, null http://hdl.handle.net/10366/166738 2025-07-31T00:02:21Z 2024-01-01T00:00:00Z

[EN] We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

2024-01-01T00:00:00Z Sharma, Puneet Ramos Calle, Higinio Kanwar, Vinay Behl, Ramandeep Rajput, Mithil http://hdl.handle.net/10366/166731 2025-07-31T00:01:51Z 2024-01-01T00:00:00Z

[EN] This paper introduces and analyzes some new highly efficient iterative procedures for approximating fixed points of contractive-type mappings. The stability, data dependence, strong convergence, and performance of the proposed schemes are addressed. Numerical examples demonstrate that the newly introduced schemes produce approximations of great accuracy and comparable to other similar robust schemes appeared in the literature. Nevertheless, all the schemes developed here are more efficient than other robust schemes used for comparison.

2024-01-01T00:00:00Z Prieto Herráez, Diego Martínez Lastras, Saray Frías-Paredes, Laura Asensio Sevilla, María Isabel González Aguilera, Diego http://hdl.handle.net/10366/163827 2025-04-30T19:48:56Z 2024-03-23T00:00:00Z

[EN]For the correct operation of the electricity system, producers must provide an estimate of the energy they are going to discharge into the system, and they must face financial penalties if their forecasts are wrong. This is especially difficult in the case of renewable energies, and in particular wind energy because of its variability and intermittency. The tool proposed allows, in a first step, to improve the prediction of wind energy to be produced and, in a second step, to optimize the offer to be presented to the electricity market, so that the overall economic performance can be improved. This tool is based on the use of public information and automatic learning and has been evaluated on a set of 30 wind farms in Spain, using their historical production data. The results indicate improvements in both the accuracy of the energy estimation and the profit obtained from the energy sold.

2024-03-23T00:00:00Z Hernández Pastora, José Luis http://hdl.handle.net/10366/162415 2025-01-25T01:02:12Z 2019-01-01T00:00:00Z

[EN]The MSA system of coordinates (Hernández-Pastora 2010 Class. Quantum Grav. 27 045006) for the MQ-solution (Hernández-Pastora and Martín 1994 Gen. Relativ. Gravit. 26 877) is proved to be the unique solution of certain partial differential equation with boundary and asymptotic conditions. Such a differential equation is derived from the orthogonality condition between two surfaces which hold a functional relationship. The obtained expressions for the MSA system recover the asymptotic expansions previously calculated (Hernández-Pastora 2010 Class. Quantum Grav. 27 045006) for those coordinates, as well as the Erez-Rosen coordinates in the spherical case. It is also shown that the event horizon of the MQ-solution can be easily obtained from those coordinates leading to already known results. But in addition, it allows us to correct a mistaken conclusion related to some bound imposed to the value of the quadrupole moment (Hernández- Pastora and Herrera 2011 Class. Quantum Grav. 28 225026). Finally, it is explored the possibility of extending this method of generalizing the Erez-Rosen coordinates to the general case of solutions with any finite number of relativistic multipole moments (RMM). It is discussed as well, the possibility of determining the Weyl moments of those solutions from their corresponding MSA coordinates, aiming to establish a relation between the uniqueness of the MSA coordinates and the solutions itself.

2019-01-01T00:00:00Z Suárez-Urango, Daniel Ospino Zúñiga, Justo Hernán Hernández, Hector Núñez, Luis http://hdl.handle.net/10366/162353 2026-01-13T10:08:07Z 2022-02-25T00:00:00Z

[EN]This paper explored the physical acceptability conditions for anisotropic matter configurations in General Relativity. The study considered a generalized polytropic equationofstate P = κργ +αρ−β foraheuristicanisotropy. We integrated the corresponding Lane–Emden equation for several hundred models and found the parameter-space portion ensuring the physical acceptability of the configurations. Polytropes based on the total energy density are more viable than those with baryonic density, and small positive local anisotropies produce acceptable models. We also found that polytropic configurations where tangential pressures are greater than radial ones are also more acceptable. Finally, convective disturbances do not generate cracking instabilities. Several models emerging from our simulations could represent candidates of astrophysical compact objects.

2022-02-25T00:00:00Z Suárez-Urango, Daniel Becerra, Laura M. Ospino Zúñiga, Justo Hernán Nuñez, Luis A. http://hdl.handle.net/10366/162350 2026-01-13T10:09:01Z 2023-11-09T00:00:00Z

[EN]We studied five methods to include anisotropy, or unequal stress distributions, in general relativistic matter configurations. We used nine acceptability conditions that the metric and physical variables must meet to determine if our models were astrophysically viable. Our analysis found the mosteffective way to introduce anisotropy while keeping a simple density profile. We also found a practical “rule of thumb” that relates the density at the boundary to the density at the centre of relativistic matter distributions. Additionally, we calculated the configuration radius and encountered that values observed by NICER for PSR J0740+6620 are consistent with several acceptable matter configurations, both isotropic and anisotropic.

2023-11-09T00:00:00Z Herrera, Luis Di Prisco, Alicia Ospino Zúñiga, Justo Hernán Carot, Jaume http://hdl.handle.net/10366/162349 2026-01-13T10:09:45Z 2023-09-15T00:00:00Z

[EN]We carry out a systematic study on the motion of test particles in the region inner to the naked singularity of a quasi-hyperbolically symmetric γ-metric. The geodesic equations are written and analyzed in detail. The obtained results are contrasted with the corresponding results obtained for the axially symmetric γ-metric and the hyperbolically symmetric black hole. As in this latter case, it is found that test particles experience a repulsive force within the horizon (naked singularity), which prevents them from reaching the center. However, in the present case, this behavior is affected by the parameter γ which measures the departure from the hyperbolical symmetry. These results are obtained for radially moving particles as well as for particles moving in the θ − r subspace. The possible relevance of these results in the explanation of extragalactic jets is revealed.

2023-09-15T00:00:00Z Herrera, Luis Di Prisco, Alicia Ospino Zúñiga, Justo Hernán http://hdl.handle.net/10366/162346 2026-01-13T10:10:37Z 2023-03-19T00:00:00Z

[EN]We search for exact analytical solutions of spherically symmetric dissipative fluid distributions satisfying the vanishing expansion condition (vanishing expansion scalar Θ). To accomplish this, we shall impose additional restrictions allowing integration of the field equations. The solutions are analyzed, and possible applications to astrophysical scenarios as well as alternative approaches to obtaining new solutions are discussed.

2023-03-19T00:00:00Z Herrera, Luis Di Prisco, Alicia Ospino Zúñiga, Justo Hernán http://hdl.handle.net/10366/162345 2026-01-13T10:11:26Z 2024-10-25T00:00:00Z

[EN]Exact solutions are presented which describe, either the evolution of fluid distributions corresponding to a ghost star (vanishing total mass), or describing the evolution of fluid distributions which attain the ghost star status at some point of their lives. The first two solutions correspond to the former case, they admit a conformal Killing vector (CKV) and describe the adiabatic evolution of a ghost star. Other two solutions corresponding to the latter case are found, which describe evolving fluid spheres absorbing energy from the outside, leading to a vanishing total mass at some point of their evolution. In this case the fluid is assumed to be expansion–free. In all four solutions the condition of vanishing complexity factor was imposed. The physical implications of the results, are discussed.

2024-10-25T00:00:00Z Herrera, Luis Di Prisco, Alicia Ospino Zúñiga, Justo Hernán http://hdl.handle.net/10366/162344 2026-01-13T10:12:07Z 2024-05-05T00:00:00Z

[EN]We explore an idea put forward many years ago by Zeldovich and Novikov concerning the existence of compact objects endowed with arbitrarily small mass. The energy density of such objects, which we call “ghost stars”, is negative in some regions of the fluid distribution, producing a vanishing total mass. Thus, the interior is matched on the boundary surface to Minkowski space–time. Some exact analytical solutions are exhibited and their properties are analyzed. Observational data that could confirm or dismiss the existence of this kind of stellar object are discussed

2024-05-05T00:00:00Z Herrera, Luis Di Prisco, Alicia Ospino Zúñiga, Justo Hernán http://hdl.handle.net/10366/162343 2026-01-13T10:12:47Z 2024-03-12T00:00:00Z

[EN]A seminumerical approach proposed many years ago for describing gravitational collapse in the post-quasi-static approximation is modified in order to avoid the numerical integration of the basic differential equations the approach is based upon. For doing that we have to impose some restrictions on the fluid distribution. More specifically, we shall assume the vanishing complexity factor condition, which allows for analytical integration of the pertinent differential equations and leads to physically interesting models. Instead, we show that neither the homologous nor the quasihomologous evolution are acceptable since they lead to geodesic fluids, which are unsuitable for being described in the post-quasi-static approximation. Also, we prove that, within this approximaCitation: Herrera, L.; Di Prisco, A.; Ospino, J. The Post-Quasi-Static Approximation: An Analytical Approach to Gravitational Collapse. Symmetry 2024, 16, 341. https:// doi.org/10.3390/sym16030341 Academic Editors: Stefano Profumo, Sergei D. Odintsov, Sunil Kumar Tripathy, Dipanjali Behera and HoomanMoradpour Received: 9 February 2024 Revised: 23 February 2024 Accepted: 6 March 2024 Published: 12 March 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). tion, adiabatic evolution also leads to geodesic fluids, and therefore, we shall consider exclusively dissipative systems. Besides the vanishing complexity factor condition, additional information is required for a full description of models. We shall propose different strategies for obtaining such an information, which are based on observables quantities (e.g., luminosity and redshift), and/or heuristic mathematical ansatz. To illustrate the method, we present two models. One model is inspired in the well-known Schwarzschild interior solution, and another one is inspired in Tolman VI solution.

2024-03-12T00:00:00Z Herrera, Luis Prisco, Alicia Di Ospino Zúñiga, Justo Hernán http://hdl.handle.net/10366/162340 2025-04-30T19:48:58Z 2024-01-03T00:00:00Z

[EN]It is shown that the evolution of an axially and reflection symmetric fluid distribution, satisfying the Tolman condition for thermal equilibrium, is not accompanied by the emission of gravitational radiation. This result, which was conjectured by Bondi many years ago, expresses the irreversibility associated to the emission of gravitational waves. The observational consequences emerging from this result are commented. The resulting models are not only nondissipative and vorticity free, but also shear-free and geodesic, furthermore all their complexity factors vanish.

2024-01-03T00:00:00Z Hernández Pastora, José Luis http://hdl.handle.net/10366/161309 2026-01-13T10:14:14Z 2020-01-01T00:00:00Z

[EN]A method to construct interior axially symmetric metrics that appropriately match with any vacuum solution of the Weyl family is developed in Hernández-Pastora et al. (Class Quantum Gravity 33:235005, 2016). It was shown, for the case of some vacuum solutions, that the simplest solution for the interior metric leads to sources with well-behaved energy conditions. Now, we integrate the field equations to obtain the interior metric functions in terms of the anisotropies and pressures of the source. As well, the compatible equations of state for these global models are calculated. The interior metric and the suitable energy–momentum tensor describing the source are constructed in terms of the exterior metric functions. At the boundary of the compact object, the behaviour of a pressure Tm, defined from the energy–momentum tensor, is shown to be related with the exterior gravitational field. This fact allows us to explore the differences arising at the matter distribution when the spherical symmetry of the global metric is dropped. Finally, an equation derived from the matching conditions is obtained which allows us to calculate the Weyl coefficients of the exterior metric as source integrals. Hence the Relativistic Multipole Moments of the global model can be expressed in terms of the matter distribution of the source.

2020-01-01T00:00:00Z