María del Pilar Rojas-Rueda

The calculation of Bernoulli and Euler numbers is based on the fact that these numbers appear in various mathematical expressions and engineering phenomena. Although computers and software programs can generate large quantities of these numbers, only a limited number are published in current references. Consequently, their use often results in unsatisfactory approximations. The objective was to calculate and evaluate the first twenty (20) Bernoulli and Euler numbers, subsequently applying them to mathematics and proposing their use in environmental engineering. There is a wide variety of relationships involving Bernoulli and Euler numbers that appear in several practical mathematical applications. In both cases, the most effective relationships used in this study are binomial expansions. By employing the appropriate and proposed equations, the Bernoulli and Euler numbers were calculated both manually and mechanically, successfully determining the number of values set as the goal. Finally, Bernoulli and Euler numbers significantly contribute to the estimation of various processes and mathematical applications, such as calculating the integrals of In (cos x) and sec(ax)/x; respectively.

Understanding the thermo-rheological properties of fruit-based beverages is critical for optimizing their industrial processing, stability, and sensory quality. This study evaluates the viscosity and flow behavior of depectinized passion fruit nectar (P. ligularis), standardized to 25° Brix, across a temperature range of 5 to 85°C. Apparent viscosity was measured using a Brookfield DV3T rotational viscometer with concentric cylinder geometry. The nectar displayed Newtonian flow behavior, maintaining consistent viscosity regardless of shear rate. Viscosity decreased significantly with rising temperature, from 0.00778 Pa·s at 5°C to 0.00018 Pa·s at 85°C. The Arrhenius model was applied to describe the temperature dependence of viscosity, yielding an activation energy (Ea) of 38.35 kJ/mol. The model fit was statistically robust (R² = 0.9913; RMSE = 0.000485 Pa·s), indicating predictable thermal behavior. Enzymatic depectinization played a key role in achieving this flow profile, eliminating structural polysaccharides that would otherwise induce non-Newtonian characteristics. These findings are industrially relevant for the design of efficient pasteurization, pumping, and packaging systems, especially for clean-label juice products. By characterizing the thermo-rheological response of this underexplored tropical nectar, this study contributes original data to the field of food engineering and supports the formulation of stable, energy-efficient fruit-based beverages.

The quality factors of prickly pear juice (Opuntia ficus indica), such as the flow index, consistency index, and yield stress index, proved to be important in the rheological characterization and acceptance of this food; reducing the challenges for technologies related to this product. The objectives were to characterize the rheology of prickly pear juice, determine the flow index, yield stress threshold, consistency index (K) and Pearson regression coefficients (r) and determination (r² × 100) for a temperature of 20 °C and concentrations of 5, 15, 25, and 40 °Brix. The methodology was based on the rheology of natural prickly pear juice, red-violet variety, mixed with 25% cold boiled water (p/p), with a pH of 3.64 and acidity of 0.85%. Measurements were carried out using a concentric cylinder viscometer, with a thermostatic bath for temperature control within a 0.5 °C range, from 15 °C to 70 °C, obtaining the shear stress (τ) corresponding to the strain rate gradients (γ). To determine the yield stress (τ₀) of the Herschel-Buckley rheological model (τ = K * γⁿ + τ₀), the procedure followed the theory of Bronshtein, et al. [1]. The flow index (n) and yield stress (τ₀) for prickly pear juice at concentrations of 5, 15, 25, and 40 °Brix were 0.5826, 0.7302, 0.7901, and 0.8611; and 58.0935, 65.2047, 73.3424, and 94.2155, respectively, which were progressive and directly proportional to the concentrations; concluding that the consistency index decreases and increases within the working concentration range, while the yield stress and flow index increase, and the regression (r) and determination (r² × 100) coefficients show a perfect correlation between shear stress (τ) and strain rate gradient (γ).

Objective. Determine was statistical mathematically the percentage (%) of prevalence of childhood caries in children between the ages of one and thirteen years as a function of diet and oral hygiene wearing the Empirical Modeling Theory of Bronstein & Semendiaev. Design. Correlational, predictive and transversal study. Participants. Patients from one year to thirteen years old were considered and distributed in four groups from one to four years old; from five to seven years old, from eight to ten years old and from eleven to thirteen years old; respectively. Twelve children have been considered in each group and we have worked with the average percentage of prevalence of each group; in which each average percentage of prevalence was contrasted with age during eight consecutive years. Main measurement. Each dispersogram showed that the first and third groups have similar behavior (sigmoidal or logistic) and groups two and four showed different behavior to groups one and three, but similar behavior between them (quadratic). Results. The statistical-mathematical analysis for groups one and three turned out tobe good, with coefficients of determination of r2=0,9752 and r2=0,9347respectively, and the relationship between time t(years) and the percentage (%) of prevalence of childhood caries (P), for children between 1 and 4 years old and 8 to 10 years old is real, there is a non-significant difference, the predictive models have high estimation, there is a "very strong correlation" between the elapsed time (t) and the percentage (%) of prevalence of childhood caries (P).Conclusions. The statistical-mathematical analysis for groups two and four proved to be good, with coefficients of determination of r2=0,9595 and r2=0,8562. respectively, and the relationship between time, t(years) and the percentage (%) of prevalence of childhood caries (P), for children between five and seven years and eleven to thirteen years, is real, there is a non-significant difference, the predictive models have high estimation, there is a "very strong correlation" between the elapsed time (t) and the percentage (%) prevalence of childhood caries (P).

Objective. Determine was mathematically modeled using the expression , which is a predictive equation. Using this model, the number of deaths due to COVID-19 worldwide was estimated.Design. Correlational, prospective, predictive and transversal study. Participans. The data on deceased individuals due to the COVID-19 disease up to November 5, 2022, was considered. Main measurement. This data was used to analyze the pandemic dispersion, which was determined to exhibit logistic sigmoidal behavior. By deriving Equation 3, the rate of deaths due to COVID-19 worldwide was calculated, obtaining the predictive model represented in Figure 3.Results. Using Equation (5), the critical time and the maximum speed and the date when the global death rate due to COVID-19 reached its maximum was July 6, 2021. The Pearson correlation coefficient between the elapsed time () and the number of deceased individuals () worldwide, based on 33 cases, was . Conclusions. This indicates that the relationship between elapsed time and the number of deceased individuals is real, with no significant difference, showing that the predictive model provides a high estimation of the correlated data.There is a "very strong correlation" between elapsed time () and the number of deceased individuals () with 87,7 % of the variance in explained by , ue to the COVID-19 disease. These models help us predict the behavior of disease like COVID-19.

La viruela del mono (mpox) es un virus de la misma familia que la viruela humana, detectada en regiones boscosas tropicales en mayo de 2022 en el Reino Unido y posteriormente en América Latina. El objetivo de la presente investigación fue estudiar, mediante modelos matemáticos, la infectividad del Monkeypox en Norteamérica en comparación epidemiológica con Sudamérica. Se ha tomado en cuenta los datos de contagiados por la viruela del mono tanto en Norteamérica y Sudamérica entre inicio de junio 2022 y fines de febrero 2023. Realizando la dispersión pandémica, se determinó que el modelo matemático de estimación es ; con el cual se estimó el número de personas contagiadas con la viruela del mono, tanto en Norteamérica como en Sudamérica. Se determinó modelo matemático de estimación de la velocidad de contagios con la viruela del mono, tanto de Norteamérica como de Sudamérica, es: . A nivel de significancia del 5 %, las varianzas de las constantes ( ) de los dos modelos son homogéneas, los contagios por la viruela del mono en Norteamérica como en Sudamérica tienen diferencia no significativa; las constantes de proporcionalidad de los modelos son iguales y existe una “correlacion muy fuerte” entre el tiempo y las personas contagiadas por la viruela del mono (mpox).

Describir la pandemia de la COVID-19 en el Perú, realizar un modelamiento estadístico matemático, determinar el tiempo crítico, la velocidad con que se desarrolló la pandemia y validar de los datos estimados; han caracterizado esta investigación; cuyo objetivo ha sido modelar y analizar los contagios por COVID-19 en el Perú, y comparar las personas contagiadas y las personas estimadas contagiadas; valorar el tiempo crítico en la que se produce la velocidad máxima de personas estimadas contagiadas y validar estadísticamente el modelo. Se ha tomado en cuenta los datos de contagios por la COVID-19 hasta el veinticuatro de febrero del 2023; llegando a determinar que describen una dispersión logística sigmoidal; suceso que fue modelado matemáticamente mediante la expresión , que es una ecuación logística predictora. Con el modelo matemático predictivo se estimó el número de personas contagiadas y su comportamiento de la COVID-19 en el Perú. De igual forma se evaluó la velocidad de las personas contagiadas con la COVID-19 en el Perú. Se estimó el tiempo crítico ( ) para la cual la velocidad de personas contagiadas fue máxima, valores que son y la velocidad máxima , respectivamente y la fecha que hubo la máxima velocidad de contagios por la COVID-19, fue el 28 de febrero del 2022. El coeficiente de correlación de Pearson para el tiempo transcurrido ( ) y el número de personas contagiadas ( ) en el Perú, por la COVID-19, basado en 37 casos, fue de ; determinando que la relación entre el tiempo y el número de contagios, es real, que el modelo predictivo tiene alta estimación de los datos correlacionados, que existe una “correlacion muy fuerte” entre el tiempo transcurrido ( ) y el número de personas contagiadas ( ) y que el 63 % de la variancia en es explicada por . Se concluye que el modelo logístico se puede aplicar con rigurosidad a fenómenos pandémicos y epidemiológicos con alta resolución y con alto grado de estimación y, se ha determinado que el coeficiente de correlación tiene una “asociación negativa muy fuerte” entre el número de contagios por la COVID-19 y el tiempo transcurrido en días.