Population growth is a topic of great interest to biologists, epidemiologists, ecologists, microbiologists and bioanalysts. Describing the dynamics of a population system through mathematical models is very useful in order to predict the behavior of the study population. Chaos theory supports studies of this type through the analysis of the logistic equation which allows observing this behavior under the variation of the constant k that represents the rate of increase in the number of times of the population values in a given time and the orbit diagram that summarizes the asymptotic behavior of all orbits in which we have values of k between zero and four. These models work with discrete time under measurement by iteration in observation and not continuously. The objective is to show the relationship of the logistic equation and the orbit diagram with the Feigenbaum constant in order to show the order that exists in the population dynamics.

Urban population growth, together with the industrial advances, leads to consider time as one of the most precious goods and therefore, an indicator of quality of life. Larger cities usually have mobility challenges due to inefficient or insufficient transport means or inadequate design of the road layout. Therefore, people may spend large portions of their daytime moving around places. It is precisely in this type of cities, where it is more complicated to find solutions that really solve the problem because it would be harder to change the already existing infrastructure. Two roads compositions, from the cities of Barcelona, Spain, and La Plata, Argentina, were selected to incorporate a fractal analysis in order to relate it with homogeneity. This was performed using a box counting methodology which determines the fractal dimension. As a complement, fractal lagunarity was calculated to get a value of the existing voids in the magnetic resonance images. The results show that the images have similar fractal dimension regarding the differences in road structure. However, the lagunarity is not the same, La Plata, Argentina, has a lower value than Barcelona, Spain, which indicates more coherence in the road structure.

The air quality index shows the pollution of the air we breathe, a very low index indicates that we can have respiratory, eye or heart conditions. An index below the minimum levels declared, requires generating strategies such as days without the presence of private vehicles, among others, to reduce pollution. The city of Medellin, Colombia, has shown very low levels in the air quality index due to several factors, mainly the emission produced by vehicles and companies. Due to this, activities have been initiated to reduce the emission of 2.5 micron and 10 microns particulate material, classified as highly harmful to health. The city installed 12 air quality monitoring stations and designed a decontamination proposal for the year 2030. Conducting a study on the persistence of particulate matter in the air according to the data recorded by the stations, will strengthen the proposed strategy or modify it to achieve the goal. In this work, the data collected by a station from July to December 2018 in which a high persistence in the presence of particulate material was analyzed.

Alzheimer’s disease is a degenerative neurological disorder that comes with the aging process characterized by cognitive, affective and behavioral deficits. a possible method that can support the diagnosis and treatment of Alzheimer’s disease is called fractal geometry, given that the brain has fractal structure, therefore, Alzheimer’s disease is an observed progressive process of alterations within the groups of topographic voxels, which means that a geometrical fractal analysis can support the diagnosis of the disease. For this purpose, a double blind clinical trial study was designed, calculating the fractal dimension and the lagunarity of magnetic resonance imaging of the cerebral cortex of patients with the neurological report of Alzheimer’s disease, implementing the box-counting method or “entropy of kolmogorov”, by means of a software with logarithm FracLac (p=1.029). As a relevant result, a characteristic fractal pattern was evidenced in the images of patients with Alzheimer’s disease, which allowed to postulate the mathematical method as a possible analysis to complement the clinical order in the diagnostic assessment.

Fractional Brownian motion allows you to see the correlation between past data increments in a shift with respect to future increments. Through this important physical property of fractional Brownian motion, it was possible to perform an analysis of the impact of suspended particles, also known as particulate matter, on the environment, which is a worldwide concern. The particulate matter produced in a coal mine is large and can be disastrous for neighboring communities, resulting in a decrease in their health and quality of life in general. In this work we analyze a time series of the values of total suspended particles registered by the “La Jagua” monitoring station located 5 km from the coal mine complex in the municipality of “La Jagua” de Ibrírico in Colombia, we do this by using the fractional Brownian motion calculated from Hurst’s coefficient to determine if there is persistence of the presence of this particulate material in the future. Initially a normality test is carried out, then the rescaled range and Hurst coefficient are computed with which it is possible to determine the increasing persistence and volatility of the series.