Journal of Mathematics Letters

In this paper, we focus on human stature, which reflects both genetic inheritance and environmental influences, particularly nutrition. This study introduces a variable-order fractional Caputo model that integrates genetic contributions, nutritional dynamics, and demographic modifiers. By allowing the fractional order α(t) to vary with age, the model captures hereditary memory and delayed responses to environmental changes. Because analytical solutions for... In this paper, we focus on human stature, which reflects both genetic inheritance and environmental influences, particularly nutrition. This study introduces a variable-order fractional Caputo model that integrates genetic contributions, nutritional dynamics, and demographic modifiers. By allowing the fractional order α(t) to vary with age, the model captures hereditary memory and delayed responses to environmental changes. Because analytical solutions for such models are rarely feasible, the predictor–corrector scheme was adopted. This method, combining an explicit Adams–Bashforth predictor with an implicit Adams–Moulton corrector, was chosen for its efficiency, stability, and adaptability to variable-order problems. Numerical simulations closely reproduce the WHO Child Growth Standards with minimal error, confirming the reliability of the approach. The study demonstrates that the strength of fractional modeling lies not only in its theoretical framework but also in the numerical method that makes it practical and applicable to pediatric endocrinology, nutrition policy, and family counseling. Read more

The number 12 is a superior highly composite number divisible by 2, 3, 4, and 6. Analyzing natural numbers through their prime factorization reveals striking patterns and symmetries. The duodecimal system, based on 12, provides a natural framework for structuring natural numbers into periods and groups with similar properties. These structures offer an intuitive perspective on the behavior of the... The number 12 is a superior highly composite number divisible by 2, 3, 4, and 6. Analyzing natural numbers through their prime factorization reveals striking patterns and symmetries. The duodecimal system, based on 12, provides a natural framework for structuring natural numbers into periods and groups with similar properties. These structures offer an intuitive perspective on the behavior of the divisor and sigma functions and on residue-class constraints relevant to Goldbach-type decompositions. Furthermore, they help to interpret observed patterns in prime gaps, accounting for the prominence of certain gaps and suggesting empirical trends. This article does not claim new theorems in analytic number theory. Its contribution is a unified mod‑12 framework that brings together several classical facts and makes certain structural regularities visually and arithmetically explicit. Read more

This article presents a mathematical model designed to simulate the impact of targeted interventions aimed at preventing HIV transmission among female sex workers (FSWs) and their clients, while also analyzing their effects on the health of the general population. The compartmental model distinguishes between high-risk populations (FSWs and their clients) and low-risk populations (sexually active men and women in the... This article presents a mathematical model designed to simulate the impact of targeted interventions aimed at preventing HIV transmission among female sex workers (FSWs) and their clients, while also analyzing their effects on the health of the general population. The compartmental model distinguishes between high-risk populations (FSWs and their clients) and low-risk populations (sexually active men and women in the general population), and links prevention efforts in high-risk groups to the evolution of the epidemic in the low-risk population. The fundamental properties of the model, such as the positivity of solutions and the boundedness of the system, have been verified, and the basic reproduction number R₀ has been calculated. Finally, the stability of the model was studied using Varga’s theorem and the Lyapunov method. Simulation results show that targeted prevention among FSWs and their clients reduces HIV incidence in the general population. This framework provides a valuable tool for guiding policymakers in the design of effective strategies to combat the epidemic, especially relevant in the context of suspension of USAID funding. Read more

A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer En≥4, the residual set ℛ(En)={En−p∣ p<En,p∈ℙ} is defined, and the universal residual set ℛE=∪​Enℛ(En) is constructed. It is shown that ℛE contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of... A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer $E_n\ge 4$, the residual set $ℛ\left(E_n\right)=\{E_n-p\mid$ $p<E_n,p\in \mathbb{P}\}$ is defined, and the universal residual set ${ℛ}_{\mathbb{E}}={{{\displaystyle \cup }}^{\text{}}}_{E_n}ℛ\left(E_n\right)$ is constructed. It is shown that ${ℛ}_{\mathbb{E}}$ contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies $G\left(E\right)\ge 2$ for all $E\ge 8$, and that the cumulative partition count satisfies ${\displaystyle \sum }_{E\le N}G\left(E\right)\gg \frac{N^2}{{\text{log}}^{4}N}$. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition $E_n=p+q$ is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer $n\ge 7$ can be expressed as $n=p+2q$, where $p,q\in \mathbb{P}$. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all $n\ge 9$. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence. Read more

Logistic growth model and its variants have been adjudged to be the most appropriate model for forecasting human population. However, in this article, we estimated the carrying capacity of Abuja using the logistic model. Then, we presented the parameters used to ascertain that the logistic model has the best fit in modelling population growth of Abuja over time. Meanwhile, a... Logistic growth model and its variants have been adjudged to be the most appropriate model for forecasting human population. However, in this article, we estimated the carrying capacity of Abuja using the logistic model. Then, we presented the parameters used to ascertain that the logistic model has the best fit in modelling population growth of Abuja over time. Meanwhile, a population growth sensitivity analysis is presented for the year 1962 to 2200.The result shows that by the year 2050, Abuja population growth rate will be out of control, if nothing substantial is implemented. Similarly, from the year 2150, the results show that stability will return again. Furthermore, the result of the error analysis conducted on the logistic model shows that Abuja has a growing population and that logistic growth model with MAPE and RMSE values of 0.98% and 7,817.07 respectively is the most accurate. The study concludes that logistic growth model with R−squared value of 0.776 has the best fit for population growth projection of Abuja. With approximate growth rate at 9.3% per annum, the projected population of Abuja will hit 30,220,701 million by the year 2039 all things being equal. Therefore, we recommend that the government should invest in massive agricultural reforms to accommodate the growing population, expand Abuja by developing its suburbs, and engage in massive reorientation of the populace on the dangers of uncontrolled births and the education of the girl child. Read more

This paper investigates the constructability of angles multiples of 3 within the framework of Euclidean geometry. It makes a significant contribution by presenting the first geometric construction for all such angles, offering a rigorous solution to a longstanding geometric problem. The paper reaffirms the efficacy of Euclidean geometry in providing precise constructions and robust proofs for these angles, demonstrating the... This paper investigates the constructability of angles multiples of 3 within the framework of Euclidean geometry. It makes a significant contribution by presenting the first geometric construction for all such angles, offering a rigorous solution to a longstanding geometric problem. The paper reaffirms the efficacy of Euclidean geometry in providing precise constructions and robust proofs for these angles, demonstrating the enduring strength of Euclidean principles from classical to modern times. The presented workflow goes beyond Euclidean geometry to examine non-Euclidean methods, particularly analytical approaches, revealing misconceptions that compromise the genetic and geometric rigor of Euclidean principles. The paper exposes incongruities when algebraic proofs related to angle constructability are applied to the Euclidean system, emphasizing the misalignment of fundamental geometric concepts. A notable result in the paper is the construction of a angle, introducing the “ angle chord” as a novel geometric property. This property challenges assumptions made by non-Euclidean methods and highlights the nuanced geometric properties crucial for rigorous constructions. The paper refutes the fallacy of relying solely on algebra for solutions to angles multiples of , emphasizing the necessity of embracing Euclidean geometry for geometric discoveries. The paper underscores the merits and resilience of Euclidean geometry, showcasing its independence and depth across historical and modern perspectives. The newly presented geometric construction not only resolves a longstanding question but also emphasizes the intrinsic strength and uniqueness of Euclidean principles in contrast to alternative methodologies. Read more