In this study, we prove the of existence of solutions of a convolution Volterra integral equation... more In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball B r belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set r M B ∈ the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.
The challenge of distinguishing genetic mutations that contribute to tumor growth is crucial in c... more The challenge of distinguishing genetic mutations that contribute to tumor growth is crucial in cancer treatment. Cancer is responsible for millions of deaths annually, hence the need for early detection of tumors to improve treatment efficacy and survival rates. However, manual classification is prone to errors and inefficiencies due to human limitations and the complexity of domain knowledge, leading to time-intensive processes. In response, machine learning models improve accuracy and efficiency for cancer prognosis and prediction. However, the lack of theoretical understanding of algorithms may limit the interpretability and applicability of results, where insights into model prediction are crucial to making informed decisions, especially in the biomedical domain. To address these challenges, our study employed four supervised machine learning algorithms, namely Support Vector Machine (SVM), Naïve Bayes (NB), Logistic Regression (LR), and Random Forest (RF). The performance of these algorithms was assessed using log-loss and misclassification rates. Logistic regression emerged as the optimal classifier with a log loss of 1.0125 and a misclassification rate of 30.97%.
Asian Research Journal of Mathematics, Dec 27, 2023
Oscillatory solutions play a pivotal role in understanding functional differential and integral e... more Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.
In algebraic field, a homomorphism is a structure-preserving map between two algebraic structures... more In algebraic field, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).These maps are continuous functions from one topological space to another. In terms of two or more maps, one map can be continuously deformed into the other and such a deformation is called a homotopy between the two functions. It is however, interesting to know that a topological space is a geometric object, and the continuous stretching and bending of the object into a new shape is a homeomorphism. That is, homeomorphism are the mappings that preserve all the topological properties of the given space. Any given pointed topological space that provides a way to determine when two or more paths, starting and ending at a fixed base point, can be continuously deformed into each other is a fundamental group. This fundamental group is a tool used for describing what a topological space looks like. It creates an algebraic image of the space using loops in the space and these loops forms a continuous deformation within a certain interval. The goal of this study is to establish that, the equivalent class 1 (S, x o) which is made up of homotopic loops is a fundamental group with respect to "•"in the interval [0,n], ∀n ∈ Z. This is of interest because it can be used as a great tool in mathematics to study many physical systems.
This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) o... more This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) on Unemployment rate (UMP) in Ghana's economy using covariance matrix and multiple regression models. The two models were examined separately on the same data of three variables and the different outputs analysed to determine the effectiveness among the two models. The analyses of the outputs highlight the significance of both predictor variables on unemployment rate in Ghana. Scatterplot and normal probability distribution (pnorm) graphs were used to analyse the normality of the predictor variables. Data on inflation rate and GDP growth spanning from 1991 to 2017 was used. The data was transformed to n X m matrix form for covariance-variance matrix analysis. The rows in the n by m data matrix were the multivariate observations on n units. Multiple regression analysis was performed on the data. Both the two methods provided the long-run effects of the two predictor variables on the unemployment rate. However, while multiple regression model could quantify the effect of each predictor variable on the predicted variable, the covariance matrix model only quantifies the relation existing between predictor variables and the predicted variable.
Advertisement involves significant amount of money and remain as one of the major issues botherin... more Advertisement involves significant amount of money and remain as one of the major issues bothering companies and other investment sectors. Advertising is a marketing strategy directed to the consumers through any media in order to present and promote product, services and any other cause. Sale of products occurs with or without advertisement and may increase or decrease depending on the market situation. The question is, is there a mathematical model that can surely confirm the effects of advertising on sales of products? The aim of this paper is to develop a first order linear differential equation model that seeks to justify the impacts of advertising on the sale of products in Ghanaian market. Two basic variables, the advertisement and sales of product were considered for the model. A system of two differential equations from the two basic variables were developed and converted to first order non-homogeneous linear differential equation. The geometrical method was employed to obtain qualitative information about the solution of the first order non-homogeneous linear differential equation. The MATLAB software package was used to plots the graphs to illustrate the behaviour of the sale of products with respect to the time of advertisement. The findings revealed that differential equation model can be used to justify the advertising impacts on the sale of products.
Ant colony optimization (ACO) has widely been applied to solve combinatorial optimization problem... more Ant colony optimization (ACO) has widely been applied to solve combinatorial optimization problems in recent years. There are few studies, however, on its convergence time, which re?ects how many iteration times ACO algorithms spend in converging to the optimal solution. This study aims at using a Max-Min Ant System (MMAS), which belongs to Ants Algorithm to obtain optimal tour of the Travelling Salesman Problem of Ghacem. The study considered a twelve city node graph (major sales point of Ghacem) with the nodes representing the twelve cities, and the edges representing the major roads linking the cities. Secondary data of the inter-city driving distances was obtained from the Ghana Highway Authority. The results showed that the objective of finding the minimum tour from the Symmetric Travelling Salesman Problem (STSP) model by using Max-Min Ants System (MMAS) Algorithm was successfully achieved. The optimal route of total cost distance was found to be 1873Km. Therefore, Ghacem coul...
Computational Journal of Mathematical and Statistical Sciences
In this paper, we introduce a new four-parameter mixture distribution called the Harmonic Mixture... more In this paper, we introduce a new four-parameter mixture distribution called the Harmonic Mixture Burr XII distribution. The proposed model can be used to model data which exhibit bimodal shapes or are heavy-tailed. Specific properties like non-central and incomplete moments, quantile function, entropy, mean and median deviation, mean residual life, moment generating function, and stressstrength reliability are derived. Maximum likelihood estimation, ordinary least squares estimation, weighted least squares estimation, Cramér-von Mises estimation, and Anderson-Darling estimation methods were used to estimate the parameters of the distribution. Simulation studies was performed to assess the estimators and the maximum likelihood estimation was adjudged the best estimator. Using three sets of lifetime data, the empirical importance of the new distribution was determined. When compared to nine (9) extensions of the Burr XII distribution, it was clear that the proposed distribution fit the data better. Using the proposed model, a log-linear regression model called the log-harmonic mixture Burr XII is proposed.
Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no excep... more Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no exception. The unavailability of jobs and its creation as being searched by persons belonging to the labour force actively looking for jobs makes the problem escalate rapidly in a growing economy. In this seven state model, we analysed into the three main economic sectors of Ghana to investigate how unemployment, employment, and newly created vacancy creation behave at equilibrium on the three economic sector levels. Moreover, we analysed how in a specific sector, the dynamics of the state variables control unemployment. Further analyses on the parameters indicated that, an increase in the rate of newly created vacancies results in a decrease in the number of unemployed persons and an increase in the number of employed. We assumed that a jobless person who is available for work but fails to make an effort to seek work is not part of the unemployed class among others. It was established that t...
International Journal of Mathematics and Mathematical Sciences
In this study, we propose a four-parameter probability distribution called the harmonic mixture F... more In this study, we propose a four-parameter probability distribution called the harmonic mixture Fréchet. Some useful expansions and statistical properties such as moments, incomplete moments, quantile functions, entropy, mean deviation, median deviation, mean residual life, moment-generating function, and stress-strength reliability are presented. Estimators for the parameters of the harmonic mixture Fréchet distribution are derived using the estimation techniques such as the maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér–von Mises estimation, and the Anderson–Darling estimation. A simulation study was conducted to assess the biases and mean square errors of the estimators. The new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution.
Generally, the constant change in demographic trends among the aged depict gradual increase in th... more Generally, the constant change in demographic trends among the aged depict gradual increase in the size of the aged population globally. The aged population is often capitalised with degenerative conditions such as chronic illness which affect their ability to function effectively and often require special support. Despite the increase in the size of the aged population and their associated degenerative conditions, very few descriptive studies on the determinants of chronic illness among the aged has been researched especially in developing countries such as Ghana and there is no compelling evidence on the association of chronic illness and its determinants. Thus, this study seeks to analyse and predict the impact of age, gender, education, marital status, Quality of Life (QoL), social cohesion, settlement and depression on chronic illness among the aged population in Ghana. From the studies a Multinomial Logit Regression (MLR) was employed to analyse the data obtained from the Ghan...
The alarming rate of emission of Carbon Dioxide (CO 2 ) into the atmosphere and its catastrophic ... more The alarming rate of emission of Carbon Dioxide (CO 2 ) into the atmosphere and its catastrophic effect on the environment, as monitored by many governmental agencies and researchers has become a source of worry for many nations and therefore needs due attention. The emissions of CO 2 in Ghana was modelled using the regressive approach to provide statistical technique for analysing the trend of CO 2 in the atmosphere. Multiple Linear Regression (MLR) model was used to determine the relationship between the various industries that contribute to CO 2 emissions in the country. Furthermore, the autoregressive approach was also used to establish the relationship within the CO 2 emissions. Based on the activities from the industry, particularly the transport sector, this work forecasted the future trend of CO 2 emissions in Ghana and recommended the usage of electric and hydrogen fuelled vehicles as the alternative to conventional fossil fuel vehicles in the future.
Unemployment is one of the major socioeconomic issues across the globe in which Ghana... more Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no exception. The unavailability of jobs and its creation as being searched by persons belonging to the labour force actively looking for jobs makes the problem escalate rapidly in a growing economy. In this seven state model, we analysed into the three main economic sectors of Ghana to investigate how unemployment, employment, and newly created vacancy creation behave at equilibrium on the three economic sector levels. Moreover, we analysed how in a specific sector, the dynamics of the state variables control unemployment. Further analyses on the parameters indicated that, an increase in the rate of newly created vacancies results in a decrease in the number of unemployed persons and an increase in the number of employed. We assumed that a jobless person who is available for work but fails to make an effort to seek work is not part of the unemployed class among others. It was established that the model has one nonnegative equilibrium point. Lastly, we analyse the impact of perturbation of some parameters on the number of the unemployed and employed persons at equilibrium.
This paper analyses Ghana’s gross domestic product using time series Autoregressive Integrated Mo... more This paper analyses Ghana’s gross domestic product using time series Autoregressive Integrated Moving Average (ARIMA). Time series analysis involves the application of statistical models to time series data and is useful for analysing the dynamics of Gross domestic product. The Ghana’s Gross domestic products (GDP) from 1980 to 2020 were obtained from the International Monetary Fund (IMF) datasets. Box Jenkins’s methodology of time series analysis was employed to analyse the data. The autocorrelation function (ACF) and partial autocorrelation function (PACF) plot suggested an Autoregression of order one AR(1). The (ARIMA) models were obtained using the minimum AIC criteria. Model diagnostics tests were performed using Ljung-Box test. The paper established that Ghana’s GDP will incline throughout the period of 2021-2025.
This paper presents the ageing process of the human body from thehomotopy conceptinthe algebraic ... more This paper presents the ageing process of the human body from thehomotopy conceptinthe algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complexconnected cylindrical shape of the body
In this study, we prove the of existence of solutions of a convolution Volterra integral equation... more In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball B r belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set r M B ∈ the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.
The challenge of distinguishing genetic mutations that contribute to tumor growth is crucial in c... more The challenge of distinguishing genetic mutations that contribute to tumor growth is crucial in cancer treatment. Cancer is responsible for millions of deaths annually, hence the need for early detection of tumors to improve treatment efficacy and survival rates. However, manual classification is prone to errors and inefficiencies due to human limitations and the complexity of domain knowledge, leading to time-intensive processes. In response, machine learning models improve accuracy and efficiency for cancer prognosis and prediction. However, the lack of theoretical understanding of algorithms may limit the interpretability and applicability of results, where insights into model prediction are crucial to making informed decisions, especially in the biomedical domain. To address these challenges, our study employed four supervised machine learning algorithms, namely Support Vector Machine (SVM), Naïve Bayes (NB), Logistic Regression (LR), and Random Forest (RF). The performance of these algorithms was assessed using log-loss and misclassification rates. Logistic regression emerged as the optimal classifier with a log loss of 1.0125 and a misclassification rate of 30.97%.
Asian Research Journal of Mathematics, Dec 27, 2023
Oscillatory solutions play a pivotal role in understanding functional differential and integral e... more Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.
In algebraic field, a homomorphism is a structure-preserving map between two algebraic structures... more In algebraic field, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).These maps are continuous functions from one topological space to another. In terms of two or more maps, one map can be continuously deformed into the other and such a deformation is called a homotopy between the two functions. It is however, interesting to know that a topological space is a geometric object, and the continuous stretching and bending of the object into a new shape is a homeomorphism. That is, homeomorphism are the mappings that preserve all the topological properties of the given space. Any given pointed topological space that provides a way to determine when two or more paths, starting and ending at a fixed base point, can be continuously deformed into each other is a fundamental group. This fundamental group is a tool used for describing what a topological space looks like. It creates an algebraic image of the space using loops in the space and these loops forms a continuous deformation within a certain interval. The goal of this study is to establish that, the equivalent class 1 (S, x o) which is made up of homotopic loops is a fundamental group with respect to "•"in the interval [0,n], ∀n ∈ Z. This is of interest because it can be used as a great tool in mathematics to study many physical systems.
This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) o... more This paper analyses the effects of Gross Domestic Product growth (GDP) and Inflation rate (INF) on Unemployment rate (UMP) in Ghana's economy using covariance matrix and multiple regression models. The two models were examined separately on the same data of three variables and the different outputs analysed to determine the effectiveness among the two models. The analyses of the outputs highlight the significance of both predictor variables on unemployment rate in Ghana. Scatterplot and normal probability distribution (pnorm) graphs were used to analyse the normality of the predictor variables. Data on inflation rate and GDP growth spanning from 1991 to 2017 was used. The data was transformed to n X m matrix form for covariance-variance matrix analysis. The rows in the n by m data matrix were the multivariate observations on n units. Multiple regression analysis was performed on the data. Both the two methods provided the long-run effects of the two predictor variables on the unemployment rate. However, while multiple regression model could quantify the effect of each predictor variable on the predicted variable, the covariance matrix model only quantifies the relation existing between predictor variables and the predicted variable.
Advertisement involves significant amount of money and remain as one of the major issues botherin... more Advertisement involves significant amount of money and remain as one of the major issues bothering companies and other investment sectors. Advertising is a marketing strategy directed to the consumers through any media in order to present and promote product, services and any other cause. Sale of products occurs with or without advertisement and may increase or decrease depending on the market situation. The question is, is there a mathematical model that can surely confirm the effects of advertising on sales of products? The aim of this paper is to develop a first order linear differential equation model that seeks to justify the impacts of advertising on the sale of products in Ghanaian market. Two basic variables, the advertisement and sales of product were considered for the model. A system of two differential equations from the two basic variables were developed and converted to first order non-homogeneous linear differential equation. The geometrical method was employed to obtain qualitative information about the solution of the first order non-homogeneous linear differential equation. The MATLAB software package was used to plots the graphs to illustrate the behaviour of the sale of products with respect to the time of advertisement. The findings revealed that differential equation model can be used to justify the advertising impacts on the sale of products.
Ant colony optimization (ACO) has widely been applied to solve combinatorial optimization problem... more Ant colony optimization (ACO) has widely been applied to solve combinatorial optimization problems in recent years. There are few studies, however, on its convergence time, which re?ects how many iteration times ACO algorithms spend in converging to the optimal solution. This study aims at using a Max-Min Ant System (MMAS), which belongs to Ants Algorithm to obtain optimal tour of the Travelling Salesman Problem of Ghacem. The study considered a twelve city node graph (major sales point of Ghacem) with the nodes representing the twelve cities, and the edges representing the major roads linking the cities. Secondary data of the inter-city driving distances was obtained from the Ghana Highway Authority. The results showed that the objective of finding the minimum tour from the Symmetric Travelling Salesman Problem (STSP) model by using Max-Min Ants System (MMAS) Algorithm was successfully achieved. The optimal route of total cost distance was found to be 1873Km. Therefore, Ghacem coul...
Computational Journal of Mathematical and Statistical Sciences
In this paper, we introduce a new four-parameter mixture distribution called the Harmonic Mixture... more In this paper, we introduce a new four-parameter mixture distribution called the Harmonic Mixture Burr XII distribution. The proposed model can be used to model data which exhibit bimodal shapes or are heavy-tailed. Specific properties like non-central and incomplete moments, quantile function, entropy, mean and median deviation, mean residual life, moment generating function, and stressstrength reliability are derived. Maximum likelihood estimation, ordinary least squares estimation, weighted least squares estimation, Cramér-von Mises estimation, and Anderson-Darling estimation methods were used to estimate the parameters of the distribution. Simulation studies was performed to assess the estimators and the maximum likelihood estimation was adjudged the best estimator. Using three sets of lifetime data, the empirical importance of the new distribution was determined. When compared to nine (9) extensions of the Burr XII distribution, it was clear that the proposed distribution fit the data better. Using the proposed model, a log-linear regression model called the log-harmonic mixture Burr XII is proposed.
Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no excep... more Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no exception. The unavailability of jobs and its creation as being searched by persons belonging to the labour force actively looking for jobs makes the problem escalate rapidly in a growing economy. In this seven state model, we analysed into the three main economic sectors of Ghana to investigate how unemployment, employment, and newly created vacancy creation behave at equilibrium on the three economic sector levels. Moreover, we analysed how in a specific sector, the dynamics of the state variables control unemployment. Further analyses on the parameters indicated that, an increase in the rate of newly created vacancies results in a decrease in the number of unemployed persons and an increase in the number of employed. We assumed that a jobless person who is available for work but fails to make an effort to seek work is not part of the unemployed class among others. It was established that t...
International Journal of Mathematics and Mathematical Sciences
In this study, we propose a four-parameter probability distribution called the harmonic mixture F... more In this study, we propose a four-parameter probability distribution called the harmonic mixture Fréchet. Some useful expansions and statistical properties such as moments, incomplete moments, quantile functions, entropy, mean deviation, median deviation, mean residual life, moment-generating function, and stress-strength reliability are presented. Estimators for the parameters of the harmonic mixture Fréchet distribution are derived using the estimation techniques such as the maximum-likelihood estimation, the ordinary least-squares estimation, the weighted least-squares estimation, the Cramér–von Mises estimation, and the Anderson–Darling estimation. A simulation study was conducted to assess the biases and mean square errors of the estimators. The new distribution was applied to three-lifetime datasets and compared with the classical Fréchet distribution and eight (8) other extensions of the Fréchet distribution.
Generally, the constant change in demographic trends among the aged depict gradual increase in th... more Generally, the constant change in demographic trends among the aged depict gradual increase in the size of the aged population globally. The aged population is often capitalised with degenerative conditions such as chronic illness which affect their ability to function effectively and often require special support. Despite the increase in the size of the aged population and their associated degenerative conditions, very few descriptive studies on the determinants of chronic illness among the aged has been researched especially in developing countries such as Ghana and there is no compelling evidence on the association of chronic illness and its determinants. Thus, this study seeks to analyse and predict the impact of age, gender, education, marital status, Quality of Life (QoL), social cohesion, settlement and depression on chronic illness among the aged population in Ghana. From the studies a Multinomial Logit Regression (MLR) was employed to analyse the data obtained from the Ghan...
The alarming rate of emission of Carbon Dioxide (CO 2 ) into the atmosphere and its catastrophic ... more The alarming rate of emission of Carbon Dioxide (CO 2 ) into the atmosphere and its catastrophic effect on the environment, as monitored by many governmental agencies and researchers has become a source of worry for many nations and therefore needs due attention. The emissions of CO 2 in Ghana was modelled using the regressive approach to provide statistical technique for analysing the trend of CO 2 in the atmosphere. Multiple Linear Regression (MLR) model was used to determine the relationship between the various industries that contribute to CO 2 emissions in the country. Furthermore, the autoregressive approach was also used to establish the relationship within the CO 2 emissions. Based on the activities from the industry, particularly the transport sector, this work forecasted the future trend of CO 2 emissions in Ghana and recommended the usage of electric and hydrogen fuelled vehicles as the alternative to conventional fossil fuel vehicles in the future.
Unemployment is one of the major socioeconomic issues across the globe in which Ghana... more Unemployment is one of the major socioeconomic issues across the globe in which Ghana is no exception. The unavailability of jobs and its creation as being searched by persons belonging to the labour force actively looking for jobs makes the problem escalate rapidly in a growing economy. In this seven state model, we analysed into the three main economic sectors of Ghana to investigate how unemployment, employment, and newly created vacancy creation behave at equilibrium on the three economic sector levels. Moreover, we analysed how in a specific sector, the dynamics of the state variables control unemployment. Further analyses on the parameters indicated that, an increase in the rate of newly created vacancies results in a decrease in the number of unemployed persons and an increase in the number of employed. We assumed that a jobless person who is available for work but fails to make an effort to seek work is not part of the unemployed class among others. It was established that the model has one nonnegative equilibrium point. Lastly, we analyse the impact of perturbation of some parameters on the number of the unemployed and employed persons at equilibrium.
This paper analyses Ghana’s gross domestic product using time series Autoregressive Integrated Mo... more This paper analyses Ghana’s gross domestic product using time series Autoregressive Integrated Moving Average (ARIMA). Time series analysis involves the application of statistical models to time series data and is useful for analysing the dynamics of Gross domestic product. The Ghana’s Gross domestic products (GDP) from 1980 to 2020 were obtained from the International Monetary Fund (IMF) datasets. Box Jenkins’s methodology of time series analysis was employed to analyse the data. The autocorrelation function (ACF) and partial autocorrelation function (PACF) plot suggested an Autoregression of order one AR(1). The (ARIMA) models were obtained using the minimum AIC criteria. Model diagnostics tests were performed using Ljung-Box test. The paper established that Ghana’s GDP will incline throughout the period of 2021-2025.
This paper presents the ageing process of the human body from thehomotopy conceptinthe algebraic ... more This paper presents the ageing process of the human body from thehomotopy conceptinthe algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complexconnected cylindrical shape of the body
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