Journal of Mathematics Letters

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

Encryption is a method of encoding data so that only authorized parties can read or access that data. With the fast improvement of digital data exchange in electronic way, Information Security is becoming much more important in data storage and transmission. In this paper, the authors work on improving the security of data using a combination of Hill cipher and... Encryption is a method of encoding data so that only authorized parties can read or access that data. With the fast improvement of digital data exchange in electronic way, Information Security is becoming much more important in data storage and transmission. In this paper, the authors work on improving the security of data using a combination of Hill cipher and a Transposition cipher (G_p) , the plaintext will be encrypted using ordinary Hill cipher technique and encrypted again using G_p cipher there by producing a more complex ciphertext that will be very difficult for cryptanalyst to intrude, and the decryption process is done in two phases, the first decryption will return the ciphertext of the ordinary Hill cipher method and the second decryption will return the original plaintext. Read more

The price of grains like maize and sorghum is subject to significant fluctuations, which can have a significant impact on a country's economy and food security. The aim of the study is to model sorghum and maize price volatility in Nigeria. The data utilized in the study was extracted from World Bank Commodity Price Data (WBCPD), 2022. The data consists... The price of grains like maize and sorghum is subject to significant fluctuations, which can have a significant impact on a country's economy and food security. The aim of the study is to model sorghum and maize price volatility in Nigeria. The data utilized in the study was extracted from World Bank Commodity Price Data (WBCPD), 2022. The data consists of monthly prices in nominal US dollars for maize and sorghum from January 1960 – August 2022. The Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models were utilized for capturing the two-grain price volatility. Two types of conditional heteroscedastic models exist, the first group uses exact functions to control the evolution of , while the second group describes with stochastic equations. It is inferred from the result that inherent uncertainties and fluctuations existed in the prices of maize and sorghum in Nigeria which implies that the price volatility is positive and statistically significant suggesting that historical information and past shocks play a crucial role in determining the volatility observed in the grains. It is recommended that the ARCH, GARCH, EGARCH, TGARCH, PARCH, CGARCH, and IGARCH models should be employed for modeling and managing the volatility of maize and sorghum prices in Nigeria. These models have shown effectiveness in capturing different aspects of volatility, including the impact of past shocks, conditional volatility, asymmetry, and other relevant factors. Read more