We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.

Abstract:
introduction. in colombia, malaria represents a serious public health problem. it is estimated that approximately 60% of the population is at risk of the disease. objective. to describe the mortality trends for malaria in colombia, from 1979 to 2008. materials and methods. a descriptive study to determine the trends of the malaria mortality was carried out. the information sources used were databases of registered deaths and population projections from 1979 to 2008 of the national statistics department. the indicator used was the mortality rate. the trend was analyzed by join point regression. results. six thousands nine hundred and sixty five deaths caused by malaria were certified for an age-adjusted rate of 0.74 deaths/100.000 inhabitants for the study period. in 74.3% of the deaths, the parasite species was not mentioned. the trend in the mortality rate showed a statistically significant decreasing behavior, which was lower from the second half of the nineties as compared with that presented in the eighties. conclusions. the magnitude of mortality by malaria in colombia is not high, in spite of the evident underreporting. a marked downward trend was observed between 1979 and 2008. the information obtained from death certificates, along with that of the public health surveillance system will allow to modify the recommendations and improve the implementation of preventive and control measures to further reduce the mortality caused by malaria.

Abstract:
Introduction. In Colombia, malaria represents a serious public health problem. It is estimated that approximately 60% of the population is at risk of the disease. Objective. To describe the mortality trends for malaria in Colombia, from 1979 to 2008. Materials and methods. A descriptive study to determine the trends of the malaria mortality was carried out. The information sources used were databases of registered deaths and population projections from 1979 to 2008 of the National Statistics Department. The indicator used was the mortality rate. The trend was analyzed by join point regression. Results. Six thousands nine hundred and sixty five deaths caused by malaria were certified for an age-adjusted rate of 0.74 deaths/100.000 inhabitants for the study period. In 74.3% of the deaths, the parasite species was not mentioned. The trend in the mortality rate showed a statistically significant decreasing behavior, which was lower from the second half of the nineties as compared with that presented in the eighties. Conclusions. The magnitude of mortality by malaria in Colombia is not high, in spite of the evident underreporting. A marked downward trend was observed between 1979 and 2008. The information obtained from death certificates, along with that of the public health surveillance system will allow to modify the recommendations and improve the implementation of preventive and control measures to further reduce the mortality caused by malaria. Introducción. En Colombia, el paludismo representa un grave problema de salud pública. Se estima que, aproximadamente, 60 % de la población se encuentra en riesgo de enfermar o de morir por esta causa. Objetivo. Describir la tendencia de la mortalidad por paludismo en Colombia desde 1979 hasta 2008. Materiales y métodos. Se llevó a cabo un estudio descriptivo para determinar la tendencia de las tasas de mortalidad. Las fuentes de información fueron las bases de datos de las defunciones registradas y de las proyecciones de población de 1979 a 2008 del Departamento Nacional de Estadística (DANE). El indicador empleado fue la tasa de mortalidad. La tendencia se analizó mediante el software de análisis de regresión de puntos de inflexión (joinpoint). Resultados. Se certificaron 6.965 muertes por paludismo para una tasa ajustada por edad de 0,74 muertes por 100.000 habitantes para el periodo estudiado. En 74,3 % de las muertes, no se especificó la especie parasitaria. Las tasas de mortalidad por paludismo presentaron una tendencia decreciente estadísticamente significativa, que fue menor a partir de la segunda mitad d

Abstract:
In this article, we present a geometric framework to study invariant sets of dynamical systems associated with differential equations. This framework is based on properties of invariant sets for an area functional. We obtain existence results for heteroclinic and periodic orbits. We also implement this approach numerically by means of the steepest descent method.

The ultimate goal of environmental impact assessment is to guarantee
that benefits generated by a development project will not cause highly negative
effects on the environment or public health. The fulfillment of this goal
depends on the willingness of proponents and society to cooperate. The
information management, its accessibility to community and the educational
level of participants are of great relevancy too. Cooperation is not always
attainable due to conflicts between individual and community interests. Conflict
leads to a variety of cooperative and non-cooperative responses, depending on
the information available to the actors. In order to capture the tendency in
which a community perceives the proposals, we introduced an information index.
We prove that computer models have a direct impact on this information index.
This computer-based approach, leads the EIA to the paradigm of adaptive
environmental assessment and management. To implement this, a system based on
artificial intelligence and game theory was used to resolve a study case of
conflict in groundwater management.

A new analysis of a previously studied traveling agent model, showed that there is a relation between the degree of homogeneity of the medium where the agents move, agent motion patterns, and the noise generated from their displacements. We proved that for a particular value of homogeneity, the system self organizes in a state where the agents carry out Lévy walks and the displacement signal corresponds to 1/f noise. Using probabilistic arguments, we conjectured that 1/f noise is a fingerprint of a statistical phase transition, from randomness (disorder) to predictability (order), and that it emerges from the contextuality nature of the system which generates it.

Abstract:
This research is concerned with new developments and practical applications of a physically-based numerical model that incorporates new approaches for a finite elements solution to the steady/transient problems of the joint ground/surface water flows. Python scripts are implemented in Geographic Information System (GIS) to store, represent and take decisions on the simulated conditions related to the water resources management at the scale of the watershed. The proposed surface-subsurface model considers surface and groundwater interactions to be 2-D horizontally distributed and depth-averaged through a diffusive wave approach for surface flood routing. Infiltration rates, overland flows and evapotranspiration processes are considered by a diffuse discharge from surface water, non-saturated subsoil and groundwater table. Recent developments also allow for the management of surface water flow control through the capacity of diversion on river beds, spillways and outflow operations of floodgates in weirs and dams of reservoirs. Practical application regards the actual hydrology of the Mero River watershed, with two important water bodies mainly concerned with the water resources management at the Cecebre Reservoir and the present flooding of a deep coal mining excavation. The MELEF model (Modèle d’éLéments Fluides, in French) was adapted and calibrated during a period of five years (2008/ 2012) with the help of hydrological parameters, registered flow rates, water levels and registered precipitation, water uses and water management operations in surface and groundwater bodies. The results predict the likely evolution of the Cecebre Reservoir, the flow rates in rivers, the flooding of the Meirama open pit and the local water balances for different hydrological components.

Abstract:
We consider a semi-linear elliptic equation on the sphere $mathbf{S}^n subset mathbb{R}^{n+1}$ with $n$ odd and subcritical nonlinearity. We show that given any positive integer $k$, if the exponent $p$ of the nonlinear term is sufficiently close to the critical Sobolev exponent $p^*$, then there exists a positive solution with $k$ peaks. Moreover, the minimum energy solutions with $k$ peaks are such that the centers of these concentrations converge as $p o p^*$ to the solution of an underlying geometrical problem, namely, arranging $k$ points on $mathbf{S}^n$ so they are as far away from each other as possible.

Abstract:
We consider the singularly perturbed Allen-Cahn equation on a strictly convex plane domain. We show that when the perturbation parameter tends to zero there are solutions having a transition layer that tends to a straight line segment. This segment can be characterized as the shortest path intersecting the boundary orthogonally at two points.

Abstract:
Alan Turing's work in Morphogenesis has received wide attention during the past 60 years. The central idea behind his theory is that two chemically interacting diffusible substances are able to generate stable spatial patterns, provided certain conditions are met. Turing's proposal has already been confirmed as a pattern formation mechanism in several chemical and biological systems and, due to their wide applicability, there is a great deal of interest in deciphering how to generate specific patterns under controlled conditions. However, techniques allowing one to predict what kind of spatial structure will emerge from Turing systems, as well as generalized reaction-diffusion systems, remain unknown. Here, we consider a generalized reaction diffusion system on a planar domain and provide an analytic criterion to determine whether spots or stripes will be formed. It is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena. This criterion is proved rigorously in some situations, generalizing well known results for the scalar equation where the pattern selection process can be understood in terms of a potential. In more complex settings it is investigated numerically. Our criterion can be applied to efficiently design Biotechnology and Developmental Biology experiments, or simplify the analysis of hypothesized morphogenetic models.