Modeling Real Phenomena Research Articles

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing.

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics.

Elucidating the Effects of Ionizing Radiation on Immune Cell Populations: A Mathematical Modeling Approach with Special Emphasis on Fractional Derivatives

Despite recent advances in the mathematical modeling of biological processes and real-world situations raised in the day-to-day life phase, some phenomena such as immune cell populations remain poorly understood. The mathematical modeling of complex phenomena such as immune cell populations using nonlinear differential equations seems to be a quite promising and appropriate tool to model such complex and nonlinear phenomena. Fractional differential equations have recently gained a significant deal of attention and demonstrated their relevance in modeling real phenomena rather than their counterpart, classical (integer) derivative differential equations. We report in this paper a mathematical approach susceptible to answering some relevant questions regarding the side effects of ionizing radiation (IR) on DNA with a particular focus on double-strand breaks (DSBs), leading to the destruction of the cell population. A theoretical elucidation of the population memory was carried out within the framework of fractional differential equations (FODEs). Using FODEs, the mathematical approach presented herein ensures connections between fractional calculus and the nonlocal feature of the fractional order of immune cell populations by taking into account the memory trace and genetic qualities that are capable of integrating all previous actions and considering the system’s long-term history. An illustration of both fractional modeling, which provides an excellent framework for the description of memory and hereditary properties of immune cell populations, is elucidated. The mathematics presented in this research hold promise for modeling real-life phenomena and paves the way for obtaining accurate model parameters resulting from the mathematical modeling. Finally, the numerical simulations are conducted for the analytical approach presented herein to elucidate the effect of various parameters that govern the influence of ionizing irradiation on DNA in immune cell populations as well as the evolution of cell population dynamics, and the results are presented using plots and contrasted with previous theoretical findings.

Approximation of the Riesz–Caputo Derivative by Cubic Splines

Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate.

The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

A New Alpha Power Weibull Model for Analyzing Time-to-Event Data: A Case Study from Football

Statistical methodologies have wider applications in exercise science, sports medicine, sports management, sports marketing, sports science, and other related sciences. These methods can be used to predict the winning probability of a team or individual in a match, the number of minutes that an individual player will spend on the ground, the number of goals to be scored by an individual player, the number of red/yellow cards that will be issued to an individual player or a team, etc. Keeping in view the importance and applicability of the statistical methodologies in sport sciences, healthcare, and other related sectors, this paper introduces a novel family of statistical models called new alpha power family of distributions. It is shown that numerous properties of the suggested method are similar to those of the new Weibull-X and exponential type distributions. Based on the novel method, a special model, namely, a new alpha power Weibull distribution, is studied. The new model is very flexible because the shape of its probability density function can either be right-skewed, decreasing, left-skewed, or increasing. Furthermore, the new distribution is also able to model real phenomena with bathtub-shaped failure rates. Finally, the usefulness/applicability of the proposed distribution is shown by analyzing the time-to-event datasets selected from different football matches during 1964–2018.

A new local non-integer derivative and its application to optimal control problems

<abstract><p>Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.</p></abstract>

The Lomax-Claim Model: Bivariate Extension and Applications to Financial Data

The uses of statistical distributions for modeling real phenomena of nature have received considerable attention in the literature. The recent studies have pointed out the potential of statistical distributions in modeling data in applied sciences, particularly in financial sciences. Among them, the two-parameter Lomax distribution is one of the prominent models that can be used quite effectively for modeling data in management sciences, banking, finance, and actuarial sciences, among others. In the present article, we introduce a new three-parameter extension of the Lomax distribution via using a class of claim distributions. The new model may be called the Lomax-Claim distribution. The parameters of the Lomax-Claim model are estimated using the maximum likelihood estimation method. The behaviors of the maximum likelihood estimators are examined by conducting a brief Monte Carlo study. The potentiality and applicability of the Lomax claim model are illustrated by analyzing a dataset taken from financial sciences representing the vehicle insurance loss data. For this dataset, the proposed model is compared with the Lomax, power Lomax, transmuted Lomax, and exponentiated Lomax distributions. To show the best fit of the competing distributions, we consider certain analytical tools such as the Anderson–Darling test statistic, Cramer–Von Mises test statistic, and Kolmogorov–Smirnov test statistic. Based on these analytical measures, we observed that the new model outperforms the competitive models. Furthermore, a bivariate extension of the proposed model called the Farlie–Gumble–Morgenstern bivariate Lomax-Claim distribution is also introduced, and different shapes for the density function are plotted. An application of the bivariate model to GDP and export of goods and services is provided.

Procesos de modelación matemática en formación de profesores de matemáticas

El presente artículo expone las generalidades de algunas experiencias en el marco de los procesos de modelación de fenómenos reales, analizando qué aspectos potencian la formación de estudiantes para profesor de Matemáticas de la Universidad Distrital Francisco José de Caldas, igualmente pretende manifestar heurísticas asociadas a la resolución de problemas en el marco de la comprensión y caracterización de algunos fenómenos en contexto, interpretando recursos y herramientas que apoyan el trabajo de los estudiantes para profesor y permiten fortalecer el desarrollo de su pensamiento matemático. Después de realizar una introducción, el artículo presenta una contextualización del estudio, donde se describe la metodología y una caracterización de los procesos de modelación matemática que promovieron el análisis de las experiencias. Finalmente se establecen algunas conclusiones asociadas con diferentes elementos presentes en las experiencias, que permiten postular aspectos de interés para la formación de profesores de matemáticas en la actualidad.

On Clarification to the Previously Published Research Articles

In the recent literature, attempts have been made to propose new statistical distributions for modeling real phenomena of nature by adding one or more additional shape parameter (s) to the distribution of baseline random variable. The major contribution of these distributions are to obtain monotonic and non-monotonic shaped failure rates. This short article, offers a clarification to the research articles proposed by Al-Kadim and Boshi [4], El-Bassiouny et al. [9] and El-Desouky et al. [10]. A brief discussion on the properties of this general class is given. A future research motivation on this subject is also provided.

On Transmuted Flexible Weibull Extension Distribution with Applications to Different Lifetime Data Sets

In this article, a three parameters transmuted version of the flexible Weibull extension distribution called transmuted flexible Weibull extension distribution is studied. The proposed model is very flexible and is able to model real phenomena with increasing, unimodal or modified unimodal failure rates. Explicit expressions for mathematical properties are derived. Maximum likelihood estimates and asymptotic confidence bounds for the unknown parameters of the model are also obtained. Four real data sets are analyzed in order to illustrate the flexibility of the proposed distribution.

Limit cycles for two families of cubic systems

In this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena. The first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of Volterra type. To prove our results we develop a new criterion on the non-existence of periodic orbits and we extend a well-known criterion on the uniqueness of limit cycles due to Kuang and Freedman. Both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools. Moreover, in both cases, we prove that when the limit cycles exist they are non-algebraic.

A Pretty Good Fit

Data analysis is not just for statistics class. Students can look at real data that arise from whatever they are studying. This is certainly true for students studying functions. Starting in algebra courses, data analysis can show students contexts in which various functions make sense, and students can use functions to model real phenomena. Although students can do this by hand, using functions to model real phenomena is practical on a regular basis only when computers and high-powered handheld technological tools take over repetitive calculations and graphing.