本論文では，近現代日本語小説を分析対象とし，社会的文体の経時的変化と個人文体の意図的な変化をめぐって統計的分析法及び機械学習法のアプローチで計量分析を行った。まず，口語体が確立されてから日本の近現代小説の文体にどのような変化が起こっているかについて考察した。そして，現代作家の水村美苗が近代文学への憧れを原点として創作した一連の作品を取り上げ，水村の意図的な文体変化について分析した。 / 博士(文化情報学) / Doctor of Culture and Information Science / 同志社大学 / Doshisha University

This work is rooted in a need for dermatologists to explore skin characteristicsin depth. The inuence of skin disease such as acne in dermal tissues is stilla complex task to assess. Among the possibilities, high frequency ultrasoundimaging is a paradigm shift to probe and characterizes upper and deep dermis.For this purpose, a cohort of 58 high-frequency 3D images has been acquiredby the French laboratory Pierre Fabre in order to study acne vulgaris disease.This common skin disorder is a societal challenge and burden aecting late adolescentsacross the world. The medical protocol developed by Pierre Fabre wasto screen a lesion every day during 9 days for dierent patients with ultrasoundimaging. The provided data features skin epidermis and dermis structure witha fantastic resolution. The strategy we led to study these data can be explainedin three steps. First, epidermis surface is detected among artifacts and noisethanks to a robust level-set algorithm. Secondly, acne spots are located on theresulting height map and associated to each other among the data by computingand thresholding a local variance. And eventually potential inammatorydermal cavities related to each lesion are geometrically and statistically characterizedin order to assess the evolution of the disease. The results presentan automatic algorithm which permits dermatologists to screen acne vulgarislesions and to characterize them in a complete data set. It can hence be a powerfultoolbox to assess the eciency of a treatment. / Detta arbete är grundat i en dermatologs behov att undersöka hudens egenskaperpå djupet. Påverkan av hudsjukdomar så som acne på dermala vävanderär fortfarande svårt att bedöma. Bland möjligheterna är högfrekvent ultraljudsavbildningett paradigmskifte för undersökning och karakterisering av övre ochdjupa dermis. I detta syfte har en kohort av 58 högfrekventa 3D bilder förvärvatsav det Franska laboratoriet Pierre Fabre för att studera sjukdomen acne vulgaris.Denna vanliga hudsjukdom är en utmaning för samhället och en bördasom påverkar de i slutet av tonåren över hela världen. Protokollet utvecklatav Pierre Fabre innebar att undersöka en lesion varje dag över 9 dagar förolika patienter med ultraljudavbildning. Den insamlade datan visar hudens epidermisoch dermis struktur med en fantastiskt hög upplösning. Strategin vianvände för att studera denna data kan förklaras i tre steg. För det första,hittas epidermis yta bland artifakter och brus tack vare en robust level-set algoritm.För det andra, acne äckar hittas på höjdkartan och associeras tillvarandra bland mätdatan genom en tröskeljämförelse över lokala variationer.Även potentiellt inammatoriska dermala hålrum relaterade till varje lesion blirgeometriskt ochj statistiskt kännetecknade för att bedöma sjukdomens förlopp.Resultaten framför en automatisk algoritm som gör det möjligt för dermatologeratt undersöka acne vulgaris lesioner och utmärka de i ett dataset. Detta kandärmed vara en kraftfull verktygslåda för att undersöka inverkan av en behandlingtill denna sjukdom.

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Par l'accumulation de mutations bénéfiques lors de cycles successifs de mutagénèse, l'évolution dirigée offre un cadre rationnel pour l'amélioration des protéines à vocation industrielle. Elle permet une exploration large de l'espace possible des séquences ainsi que leurs capacités fonctionnelles. Elle est cependant lourde à mettre en oeuvre et nécessite des moyens importants. Des approches in silico font usage d'un jeu minimal de données expérimentales et utilisent la modélisation statistique combinée à des algorithmes d'apprentissage machine. Elles ont été développées pour explorer de façon heuristique l'espace possible des séquences et de la fitness et d'identifier les mutations et interactions entre résidus les plus intéressantes. C'est l'objet de cette thèse qui explore la construction et l'application de modèles statistiques s'appuyant sur des jeux minimaux de données expérimentales pour relier fitness, ou activité, à la séquence biologique des variants. L'étude s'articule autour d'un choix crucial d'une méthode de numérisation, de descripteurs de la séquence et de méthodes de régression. La méthode ProSAR de R. Fox (2005) et les limites de son applicabilité sur des jeux de données expérimentales ont été étudiées. De nouvelles méthodes ont aussi été développées, prenant en compte les propriétés physico-chimiques des acides aminés et leurs périodicités. Elle a permis de découvrir de nouveaux descripteurs reliant la séquence à l'activité et propose des approches innovantes qui ont la capacité de traiter des cadres biologiques très divers, même lorsque peu de données biologiques sont disponibles. / Via the accumulation of beneficial mutations through successive rounds of mutations, directed evolution offers a rational framework for the amelioration of protein of industrial interest. It enables the large exploration of the sequence space and fitness. However, they are wet-lab intensive and may reveal to be time consuming and costly. In silico approaches using minimal sets of experimental data and statistical models combined with machine learning algorithms have been developed to explore heuristically the sequence space and to identify the effect of the potential epistatic interactions between residues on protein fitness. This work focused on the construction and application of statistical models relying on minimal experimental datasets to study protein sequence to activity relationships (ProSAR). In particular, the choices of appropriate numerical encoding methods, of descriptors extracted from protein sequences and of regression methods were investigated. The original ProSAR method from R. Fox (2005) and the limits of its applicability on experimental datasets have been studied. New methods that consider physico-chemical features of amino acids and their periodicities have been explored. This study unveils novel descriptors of the sequence-activity relationship and provides innovative approaches that can deal with very diverse biological datasets, even when few biological data are available.

L’ethnomusicologie est l’étude de la musique en mettant l’accent sur les aspects culturels, sociaux, matériels, cognitifs et/ou biologiques. Ce sujet de thèse, motivé par Pr. Marc Chemillier, ethnomusicologue au laboratoire CAMS-EHESS, traite du développement d’un système automatique de transcription dédié aux répertoires de musique de la cithare marovany de Madagascar. Ces répertoires sont transmis oralement, résultant d’un processus de mémorisation/ transformation de motifs musicaux de base. Ces motifs sont un patrimoine culturel important du pays, et évoluent en permanence sous l’influence d’autres pratiques et genres musicaux. Les études ethnomusicologiques actuelles visent à comprendre l’évolution du répertoire traditionnel, et de préserver ce patrimoine. Pour servir cette cause, notre travail consiste à fournir des outils informatiques d’analyse musicale pour organiser et structurer des enregistrements audio de cet instrument. La transcription automatique de musique consiste à estimer les notes d’un enregistrement à travers les trois attributs : temps de début, hauteur et durée de note. Notre travail sur cette thématique repose sur l’incorporation de connaissances musicales a priori dans les systèmes informatiques. Une première étape de cette thèse fût donc de générer cette connaissance et de la formaliser en vue de cette incorporation. Cette connaissance explorer les caractéristiques multi-modales du signal musical, incluant le timbre, le langage musical et les techniques de jeu. La recherche effectée dans cette thèse se distingue en deux axes : un premier plus appliqué, consistant à développer un système de transcription de musique dédié à la marovany, et un second plus fondamental, consistant à fournir une analyse plus approfondie des contributions de la connaissance dans la transcription automatique de musique. Notre premier axe de recherche requiert une précision de transcription très bonne (c.a.d. une F-measure supérieure à 95 % avec des tolérances d’erreur standardes) pour faire office de supports analytiques dans des études musicologiques. Pour cela, nous utilisons une technologie de captation multicanale appliquée aux instruments à cordes pincées. Les systèmes développés à partir de cette technologie utilisent un capteur par corde, permettant de décomposer un signal polyphonique en une somme de signaux monophoniques respectifs à chaque corde, ce qui simplifie grandement la tâche de transcription. Différents types de capteurs (optiques, piézoélectriques, électromagnétiques) ont été testés. Après expérimentation, les capteurs piézoélectriques, bien qu’invasifs, se sont avérés avoir les meilleurs rapports signal-sur-bruit et séparabilité inter-capteurs. Cette technologie a aussi permis le développement d’une base de données dite “ground truth" (vérité de terrain), indispensable pour l’évaluation quantitative des systèmes de transcription de musique. Notre second axe de recherche propose des investigations plus approfondies concernant l’incorporation de connaissance a priori dans les systèmes automatiques de transcription de musique. Deux méthodes statistiques ont été utilisées comme socle théorique, à savoir le PLCA (Probabilistic Latent Component Analysis) pour l’estimation multi-pitch et le HMM (Hidden Markov Models). / Ethnomusicology is the study of musics around the world that emphasize their cultural, social, material, cognitive and/or biological. This PhD sub- ject, initiated by Pr. Marc CHEMILLIER, ethnomusicolog at the laboratory CAMS-EHESS, deals with the development of an automatic transcription system dedicated to the repertoires of the traditional marovany zither from Madagascar. These repertoires are orally transmitted, resulting from a pro- cess of memorization/transformation of original base musical motives. These motives represent an important culture patrimony, and are evolving contin- ually under the inuences of other musical practices and genres mainly due to globalization. Current ethnomusicological studies aim at understanding the evolution of the traditional repertoire through the transformation of its original base motives, and preserving this patrimony. Our objectives serve this cause by providing computational tools of musical analysis to organize and structure audio recordings of this instrument. Automatic Music Transcription (AMT) consists in automatically estimating the notes in a recording, through three attributes: onset time, duration and pitch. On the long range, AMT systems, with the purpose of retrieving meaningful information from complex audio, could be used in a variety of user scenarios such as searching and organizing music collections with barely any human labor. One common denominator of our diferent approaches to the task of AMT lays in the use of explicit music-related prior knowledge in our computational systems. A step of this PhD thesis was then to develop tools to generate automatically this information. We chose not to restrict ourselves to a speciprior knowledge class, and rather explore the multi-modal characteristics of musical signals, including both timbre (i.e. modeling of the generic \morphological" features of the sound related to the physics of an instrument, e.g. intermodulation, sympathetic resonances, inharmonicity) and musicological (e.g. harmonic transition, playing dynamics, tempo and rhythm) classes. This prior knowledge can then be used in com- putational systems of transcriptions. The research work on AMT performed in this PhD can be divided into a more \applied research" (axis 1), with the development of ready-to-use operational transcription tools meeting the cur- rent needs of ethnomusicologs to get reliable automatic transcriptions, and a more \basic research" (axis 2), providing deeper insight into the functioning of these tools. Our axis of research requires a transcription accuracy high enough 1 (i.e. average F-measure superior to 95 % with standard error tolerances) to provide analytical supports for musicological studies. Despite a large enthusiasm for AMT challenges, and several audio-to-MIDI converters available commercially, perfect polyphonic AMT systems are out of reach of today's al- gorithms. In this PhD, we explore the use of multichannel capturing sensory systems for AMT of several acoustic plucked string instruments, including the following traditional African zithers: the marovany (Madagascar), the Mvet (Cameroun), the N'Goni (Mali). These systems use multiple string- dependent sensors to retrieve discriminatingly some physical features of their vibrations. For the AMT task, such a system has an obvious advantage in this application, as it allows breaking down a polyphonic musical signal into the sum of monophonic signals respective to each string.

Este trabalho teve como objetivo o estudo experimental de lubrificação e resfriamento no torneamento duro de uma superfície funcional em componentes de grande porte, com aplicação no chão de fábrica. Foram comparadas as técnicas de mínima quantidade de lubrificante e resfriamento abundante, utilizando o método de resposta de superfície. O planejamento tridimensional utilizado foi de dois níveis, adicionado de pontos axiais, ponto central e de réplica. Foram variados a velocidade periférica da peça, o avanço da ferramenta de usinagem e a profundidade de usinagem. Após a execução do experimento, a integridade da superfície foi analisada pela medição da tensão residual, da espessura de possíveis modificações microestruturais e pela medição dos parâmetros estatísticos Sa, Sq e Sz. As medidas foram então ajustadas em modelos matemáticos, otimizadas e por fim comparadas. A análise dos resultados sinalizou que os modelos ajustados para ambos os métodos foram capazes de explicar satisfatoriamente o comportamento das variáveis de resposta. Ainda, a partir da função desirability, foi possível estimar valores ótimos com qualidade equivalente entre os métodos. Tanto para o emprego com mínimas quantidades quanto para resfriamento abundante foram registradas tensões circunferenciais altamente compressivas. Nas condições ótimas, quando empregadas mínimas quantidades de lubrificante, a superfície apresentou valores de tensões residuais 37% maiores em comparação ao obtido quando empregado o resfriamento abundante. Ambos os métodos produziram tanto superfícies livres de modificações microestruturais significativas, como também superfícies com modificações microstruturais significativas. Porém, quando detectadas, as modificações se mostraram muito reduzidas, com espessura de até 2,35 &#956m. Nas condições ótimas, quando empregado mínimas quantidades de lubrificante, a espessura da modificação foi 74% menor em comparação ao obtido quando empregado o resfriamento abundante. Os parâmetros estatísticos sugeriram alguma vantagem da aplicação de mínimas quantidades de lubrificante. Nas condições ótimas, a aplicação MQL apresentou melhor rugosidade Sa, em 47%, e Sz, em 11%. Porém, o desvio-padrão Sq da superfície apresentou valor 12% maior ao resfriamento abundante. / This work aimed to conduct a study of cooling and lubrication in hard turning of functional surface of large components. It was compared the performance of minimum quantity lubrication and abundant cooling through the application of response surface methodology. The activities were developed in shop floor application. It was designed a three-dimensional experiment with two levels, added by axial points, center point and one center replicate. The input variables were the peripheral velocity of the workpiece, the feed of the cutting tool and the depth of cut of machining. After the process were analyzed the surface integrity through the circumferential residual stress, possibly microstructure modified layers and statistical parameters such Sa, Sq and Sz. Then, the measurements were adjusted in mathematical models, optimized and compared. The analysis of the results indicated that the adjusted models for both methods were capable of explaining satisfactorily the behavior of the response variables. Also, the use of the desirability function allowed to predict optimal values with equivalent quality between the methods. The minimal quantities, as well the abundant cooling, produced circumferential residual stresses highly compressive. On optimal conditions, the MQL presented residual stresses 37% lower the abundant cooling. Both methods produced surfaces free of significant altered layers as well surface containing significant surface altered layer. However, when detected, the altered surface layer was very thin, with thickness up to 2,35 &#956m. On optimal conditions, the MQL altered layer was 74% lower the abundant cooling. The statistical parameters indicated some advantage on the application of MQL. On optimal conditions, the minimal quantities presented better Sa roughness in 47%, and Sz, in 11%. However, the standard deviation Sq of the surface presented a value 12% higher the abundant cooling.

[pt] O objetivo desta pesquisa é propor um modelo estatístico que possa estimar a probabilidade de ocorrência de falências ou concordatas em empresas de Internet. Após as recentes e drásticas perdas de capital em investimentos nas empresas desta nova indústria, instituições financeiras, pessoas físicas e todos os investidores desejam ter o conhecimento da real situação financeira das empresas denominadas pontocom. Esta pesquisa selecionou empresas norte-americanas que pediram falência ou concordata nas Cortes Norte-Americanas de Falências,entre 1999 e 2001, e empresas que não o fizeram, por amostragem de conveniência, que possuem ações listadas em bolsa e operam no e-commerce, isto é que vendem seus produtos ou serviços através da Internet. Utilizou, ainda, as demonstrações financeiras destas empresas para identificar, por intermédio de um teste T de amostras independentes, as variáveis mais significantes na discriminação dos dois grupos de empresas observados na amostra: o de empresas falidas e o de não-falidas. Analisadas as distribuições estatísticas das variáveis,o modelo de regressão logística demonstra ser o mais apropriado à pesquisa, por não possuir a premissa de normalidade multivariada. A conclusão final da pesquisa é a proposição de um modelo estatístico que indica a probabilidade de uma empresa de Internet falir ou não, com índice R2 de Nagelkerke de 0,887, percentual máximo de acerto na classificação de 97,4 por cento e que utiliza ainda não utilizadas em pesquisas anteriores similares. / [en] The objective of this research is to propose a statistical model that could estimate the probability of occurrence of bankruptcy for Internet companies. After the recent and drastic losses of investment capital in companies in this new sector of the economy, financial institutions, individuals and all investors wish to know the real financial position of these companies called dotcom. This research selected American companies that have filed a petition under the United States Bankruptcy Code, between 1999 and 2001, and companies which have not done it, by convenience sampling, that list their shares on stock markets and operate in e-commerce, i.e. companies that sell their products or services through the Internet. The financial statements of these companies were also used to identify,by analyzing a T test of independent samples, the most significant variables for discriminating the two observed groups in the sample: the bankrupt and the nonbankrupt companies. After analyzing the variables statistical distributions, a logistic regression model revealed to be the more appropriate for the research, for not having the multivariate normality assumption. The conclusion of this research proposes a statistical model which indicates the probability of an Internet company becoming bankrupt or not, with a Nagelkerke R Squared of 0,887, and an overall percentage of correct prediction of 97,4 percent. The model uses several variables not previously included in similar previous financial difficulties prediction models.

Este trabalho teve como objetivo o estudo experimental de lubrificação e resfriamento no torneamento duro de uma superfície funcional em componentes de grande porte, com aplicação no chão de fábrica. Foram comparadas as técnicas de mínima quantidade de lubrificante e resfriamento abundante, utilizando o método de resposta de superfície. O planejamento tridimensional utilizado foi de dois níveis, adicionado de pontos axiais, ponto central e de réplica. Foram variados a velocidade periférica da peça, o avanço da ferramenta de usinagem e a profundidade de usinagem. Após a execução do experimento, a integridade da superfície foi analisada pela medição da tensão residual, da espessura de possíveis modificações microestruturais e pela medição dos parâmetros estatísticos Sa, Sq e Sz. As medidas foram então ajustadas em modelos matemáticos, otimizadas e por fim comparadas. A análise dos resultados sinalizou que os modelos ajustados para ambos os métodos foram capazes de explicar satisfatoriamente o comportamento das variáveis de resposta. Ainda, a partir da função desirability, foi possível estimar valores ótimos com qualidade equivalente entre os métodos. Tanto para o emprego com mínimas quantidades quanto para resfriamento abundante foram registradas tensões circunferenciais altamente compressivas. Nas condições ótimas, quando empregadas mínimas quantidades de lubrificante, a superfície apresentou valores de tensões residuais 37% maiores em comparação ao obtido quando empregado o resfriamento abundante. Ambos os métodos produziram tanto superfícies livres de modificações microestruturais significativas, como também superfícies com modificações microstruturais significativas. Porém, quando detectadas, as modificações se mostraram muito reduzidas, com espessura de até 2,35 &#956m. Nas condições ótimas, quando empregado mínimas quantidades de lubrificante, a espessura da modificação foi 74% menor em comparação ao obtido quando empregado o resfriamento abundante. Os parâmetros estatísticos sugeriram alguma vantagem da aplicação de mínimas quantidades de lubrificante. Nas condições ótimas, a aplicação MQL apresentou melhor rugosidade Sa, em 47%, e Sz, em 11%. Porém, o desvio-padrão Sq da superfície apresentou valor 12% maior ao resfriamento abundante. / This work aimed to conduct a study of cooling and lubrication in hard turning of functional surface of large components. It was compared the performance of minimum quantity lubrication and abundant cooling through the application of response surface methodology. The activities were developed in shop floor application. It was designed a three-dimensional experiment with two levels, added by axial points, center point and one center replicate. The input variables were the peripheral velocity of the workpiece, the feed of the cutting tool and the depth of cut of machining. After the process were analyzed the surface integrity through the circumferential residual stress, possibly microstructure modified layers and statistical parameters such Sa, Sq and Sz. Then, the measurements were adjusted in mathematical models, optimized and compared. The analysis of the results indicated that the adjusted models for both methods were capable of explaining satisfactorily the behavior of the response variables. Also, the use of the desirability function allowed to predict optimal values with equivalent quality between the methods. The minimal quantities, as well the abundant cooling, produced circumferential residual stresses highly compressive. On optimal conditions, the MQL presented residual stresses 37% lower the abundant cooling. Both methods produced surfaces free of significant altered layers as well surface containing significant surface altered layer. However, when detected, the altered surface layer was very thin, with thickness up to 2,35 &#956m. On optimal conditions, the MQL altered layer was 74% lower the abundant cooling. The statistical parameters indicated some advantage on the application of MQL. On optimal conditions, the minimal quantities presented better Sa roughness in 47%, and Sz, in 11%. However, the standard deviation Sq of the surface presented a value 12% higher the abundant cooling.

Despite recent advances in statistics, artificial neural network theory, and machine learning, nonlinear function estimation in high-dimensional space remains a nontrivial problem. As the response surface becomes more complicated and the dimensions of the input data increase, the dreaded "curse of dimensionality" takes hold, rendering the best of function approximation methods ineffective. This thesis takes a novel approach to solving the high-dimensional function estimation problem. In this work, we propose and develop two distinct parametric projection pursuit learning networks with wide-ranging applicability. Included in this work is a discussion of the choice of basis functions used as well as a description of the optimization schemes utilized to find the parameters that enable each network to best approximate a response surface. The essence of these new modeling methodologies is to approximate functions via the superposition of a series of piecewise one-dimensional models that are fit to specific directions, called projection directions. The key to the effectiveness of each model lies in its ability to find efficient projections for reducing the dimensionality of the input space to best fit an underlying response surface. Moreover, each method is capable of effectively selecting appropriate projections from the input data in the presence of relatively high levels of noise. This is accomplished by rigorously examining the theoretical conditions for approximating each solution space and taking full advantage of the principles of optimization to construct a pair of algorithms, each capable of effectively modeling high-dimensional nonlinear response surfaces to a higher degree of accuracy than previously possible.

Statistical analysis and modeling are useful for understanding the behavior of different phenomena. In this study we will focus on two areas of applications: Global warming and cancer research. Global Warming is one of the major environmental challenge people face nowadays and cancer is one of the major health problem that people need to solve. For Global Warming, we are interest to do research on two major contributable variables: Carbon dioxide (CO2) and atmosphere temperature. We will model carbon dioxide in the atmosphere data with a system of differential equations. We will develop a differential equation for each of six attributable variables that constitute CO2 in the atmosphere and a differential system of CO2 in the atmosphere. We are using real historical data on the subject phenomenon to develop the analytical form of the equations. We will evaluate the quality of the developed model by utilizing a retrofitting process. Having such an analytical system, we can obtain good estimates of the rate of change of CO2 in the atmosphere, individually and cumulatively as a function of time for near and far target times. Such information is quite useful in strategic planning of the subject matter. We will develop a statistical model taking into consideration all the attributable variables that have been identified and their corresponding response of the amount of CO2 in the atmosphere in the continental United States. The development of the statistical model that includes interactions and higher order entities, in addition to individual contributions to CO2 in the atmosphere, are included in the present study. The proposed model has been statistically evaluated and produces accurate predictions for a given set of the attributable variables. Furthermore, we rank the attributable variables with respect to their significant contribution to CO2 in the atmosphere. For Cancer Research, the object of the study is to probabilistically evaluate commonly used methods to perform survival analysis of medical patients. Our study includes evaluation of parametric, semi-parametric and nonparametric analysis of probability survival models. We will evaluate the popular Kaplan-Meier (KM), the Cox Proportional Hazard (Cox PH), and Kernel density (KD) models using both Monte Carlo simulation and using actual breast cancer data. The first part of the evaluation will be based on how these methods measure up to parametric analysis and the second part using actual cancer data. As expected, the parametric survival analysis when applicable gives the best results followed by the not commonly used nonparametric Kernel density approach for both evaluations using simulation and actual cancer data. We will develop a statistical model for breast cancer tumor size prediction for United States patients based on real uncensored data. When we simulate breast cancer tumor size, most of time these tumor sizes are randomly generated. We want to construct a statistical model to generate these tumor sizes as close as possible to the real patients' data given other related information. We accomplish the objective by developing a high quality statistical model that identifies the significant attributable variables and interactions. We rank these contributing entities according to their percentage contribution to breast cancer tumor growth. This proposed statistical model can also be used to conduct surface response analysis to identify the necessary restrictions on the significant attributable variables and their interactions to minimize the size of the breast tumor. We will utilize the Power Law process, also known as Non-homogenous Poisson Process and Weibull Process to evaluate the effectiveness of a given treatment for Stage I & II Ductal breast cancer patients. We utilize the shape parameter of the intensity function to evaluate the behavior of a given treatment with respect to its effectiveness. We will develop a differential equation that will characterize the behavior of the tumor as a function of time. Having such a differential equation, the solution of which once plotted will identify the rate of change of tumor size as a function of age. The structure of the differential equation consists of the significant attributable variables and their interactions to the growth of breast cancer tumor. Once we have developed the differential equations and its solution, we proceed to validate the quality of the proposed differential equations and its usefulness.