BackgroundFractal geometry has been the basis for the development of a diagnosis of preneoplastic and neoplastic cells that clears up the undetermination of the atypical squamous cells of undetermined significance (ASCUS).MethodsPictures of 40 cervix cytology samples diagnosed with conventional parameters were taken. A blind study was developed in which the clinic diagnosis of 10 normal cells, 10 ASCUS, 10 L-SIL and 10 H-SIL was masked. Cellular nucleus and cytoplasm were evaluated in the generalized Box-Counting space, calculating the fractal dimension and number of spaces occupied by the frontier of each object. Further, number of pixels occupied by surface of each object was calculated. Later, the mathematical features of the measures were studied to establish differences or equalities useful for diagnostic application. Finally, the sensibility, specificity, negative likelihood ratio and diagnostic concordance with Kappa coefficient were calculated.ResultsSimultaneous measures of the nuclear surface and the subtraction between the boundaries of cytoplasm and nucleus, lead to differentiate normality, L-SIL and H-SIL. Normality shows values less than or equal to 735 in nucleus surface and values greater or equal to 161 in cytoplasm-nucleus subtraction. L-SIL cells exhibit a nucleus surface with values greater than or equal to 972 and a subtraction between nucleus-cytoplasm higher to 130. L-SIL cells show cytoplasm-nucleus values less than 120. The rank between 120–130 in cytoplasm-nucleus subtraction corresponds to evolution between L-SIL and H-SIL. Sensibility and specificity values were 100%, the negative likelihood ratio was zero and Kappa coefficient was equal to 1.ConclusionsA new diagnostic methodology of clinic applicability was developed based on fractal and euclidean geometry, which is useful for evaluation of cervix cytology.

It has been shown that the variability of the marginal capital product ratiois what determines the growth rate of economies according to Harrod's model; therefore, it is important to predict its behavior. This paper develops a methodology based on probabilistic random walk to predict the annual marginal capital product ratio of Colombia for the year 2017 based on information from the Penn World Table database from 1993 to 2016. It was found that this variable has underlying mathematical orders, which allows its prediction with an accuracy of 97.69%.

Background: Flow cytometry evaluates the number CD4-Positive T-Lymphocytes in patients infected with HIV/AIDS in anti-retroviral management, which orientates therapies towards different targets. Previously, a methodology was designed based on probability and set theories from leukocyte and lymphocyte counts of complete blood count, although predictions in time were not developed, which is why is wanted to establish a methodology of clinical applicability to temporarily forecast the values of CD4+ greater than 500, between 200 and 500 and lesser than 200 from the values of CD4+ and leukocytes of each patient. Methods: From sequential counts of CD4+ and leukocytes of 200 cases, the registries of 10 prototypical patients were observed to establish predictive patterns, and then these patterns were are applied to the remaining patients in a blind study, finding the probability of success of the methodology as well as sensitivity and specificity values. Results: 5 patterns were found with percentages greater than 99% of predictive accuracy for the distinct conditions of the methodology, with values of sensitivity and specificity of 99%. Conclusions: through a mathematical theoretical simplification, a temporal self-organization in the sequence of measurements of leukocytes and CD4+ lymphocytes were established, highlighting the loading of probability in the dynamic of CD4+ counts, useful to conduct more appropriate following ups of patients in anti-retroviral management.

Introducción: Anteriormente se desarrolló una ley físico-matemática para la evaluación de registros electrocardiográficos continuos y Holters con la cual se dedujo la totalidad atractores cardíacos y se diferenció normalidad, estados patológicos y evolución entre estados. Método: Fueron tomadas 200 dinámicas cardiacas, 150 con diferentes tipos de patologías cardiacas y 50 normales, a las cuales se aplicó la ley exponencial, en 18 y 21 horas. Para ello, se simuló una secuencia de frecuencias cardiacas, con la cual fue construido el atractor caótico. A continuación, se determinó el diagnóstico matemático a partir de la ley, con base en la ocupación espacial del atractor y se calculó la dimensión fractal. Finalmente, se realizó la validación estadística del método matemático en 18 horas frente al Gold Standard. Resultados: Sujetos con dinámicas cardiacas caóticas normales presentaron valores en la rejilla Kp entre 205 y 384, mientras que los sujetos con dinámicas patológicas presentaron valores entre 61 y 191 en 18 horas.  La evaluación de la concordancia entre el diagnóstico matemático en 18 horas y la evaluación convencional tomada Gold Standard, dio como resultado valores de sensibilidad y especificidad de 100% y un coeficiente Kappa de 1. Conclusión: Se confirmó la capacidad clínica de la ley para diagnosticar de forma objetiva y reproducible en 18 horas.