Research in Basic Sciences: Results of some math and physics projects

Desde nuestra Facultad se presentan trabajos resultados de investigación que le aportan ignificativamente a las aplicaciones de las ciencias básicas y que son el resultado del compromiso y la dedicación de nuestros docentes investigadores, quienes trabajan para abordar problemas complejos y generar soluciones innovadoras y sostenibles. En el primer capítulo se presenta un estudio de las propiedades en nuevos materiales, en este caso del crecimiento de películas delgadas de grafeno en sustrato de silicio utilizando el método de deposición por láser pulsado (DLP) donde se utilizaron 6 técnicas de caracterización (Espectroscopía Raman, IRTF, MEB, SED, MFA, MACA) y como conclusión se tiene que las muestras similares al grafeno presentan propiedades de hidrofobicidad y las muestras de carbono amorfo tienen propiedades hidrofílicas. En el segundo capítulo se presenta un modelamiento matemático y una discusión sobre la existencia de soluciones de segunda clase en una ecuación de Tricomi, se plantea la existencia de soluciones periódicas de una generalización de la ecuación de Tricomi, el estudio se realiza mediante métodos perturbativos; de igual forma, se usa la función de Melnikov para encontrar las condiciones bajo las cuales se conservan las curvas homoclínicas. En el tercer capítulo se plantea una metodología de enseñanza-aprendizaje para enseñar la geometría descriptiva básica y la resolución de problemas a través del software Autocad.Publicación financiada con recursos de la Vicerrectoría de Investigaciones , Innovación y Extensión de la Universidad Tecnológica de PereiraVicerrectoría de Investigaciones , Innovación y Extensión de la Universidad Tecnológica de PereiraCONTENIDO Introducción ..................................................................................................................5 CAPÍTULO UNO Study of graphene growth onto silicon substrates by pulsed laser deposition method/ Estudio del crecimiento de grafeno en sustratos de silicio usando el método de deposición por láser pulsado ......................................................................................9 Henry Riascos Landázani - Juan David Lopez Vargas CAPÍTULO DOS A note about the existence of second-kind periodic solutions in tricomi’s equations/ Una nota sobre la existencia de soluciones de segunda clase en una ecuación de tricomi ......................................................................................................................35 Daniel Cortés Zapata - Alexander Gutiérrez Gutiérrez CAPÍTULO TRES Basic descriptive geometry with AUTOCAD/ Geometría descriptiva básica con AUTOCAD.............................................................49 Alberto García Lópe

Steepest Descent for Partial Differential Equations of Mixed Type

The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u) = 1/2 II T(u) II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type). Finally, the graphical outputs on some test boundary conditions are presented in the table of illustrations

Price feedback and hybrid diffusions in finance

It is well-known that the probabilistic behaviour of financial asset returns is not captured well by the classical Black-Scholes model. The true behaviour will never be perfectly captured in any model, but insight is continually being obtained into our understanding of more sophisticated and realistic models. Much research has been published recently exploring the use of L\'{e}vy process models, which maintain the original \emph{independent stationary increments} assumption present in the Black-Scholes model, but incorporate jumps in the modelling. This investigation seeks to motivate a new class of models, throwing out the stationary increments hypothesis. We argue that certain techniques of trading decision-making are not independent of previous price movements, and the returns, being driven by the trade order flow, will reflect that. From here, we develop two particular such models, which are both diffusion models, and study them for their probabilistic behaviour. The first of these models is a hybrid of the arithmetic and geometric Brownian motions, which has transition probabilities expressible in terms of a spectral expansion involving Legendre functions. The second is a hybrid of the arithmetic Brownian motion and the Cox-Ingersoll-Ross process, and its spectral expansions involve the confluent hypergeometric functions. Having developed these expressions in sufficient detail to do so, we consider the calculation of value-at-risk and expected shortfall in these two models

Transonic flow studies

Major emphasis was on the design of shock free airfoils with applications to general aviation. Unsteady flow, transonic flow, and shock wave formation were examined

Adaptive weights smoothing with applications to image segmentation

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data-points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to Magnetic Resonance Imaging