Abstract

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 R<SUB>0</SUB> 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.

Abstract

A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2265;</mo><mn>4</mn></mrow></semantics></math>, the residual set <math><semantics><mrow><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>&#x007B;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>&#x2212;</mo><mi>p</mi><mo>&#x2223;</mo></mrow></semantics></math> <math><semantics><mrow><mi>p</mi><mo>&#x003C;</mo><msub><mi>E</mi><mi>n</mi></msub><mo>,</mo><mi>p</mi><mo>&#x2208;</mo><mi>&#x2119;</mi><mo>&#x007D;</mo></mrow></semantics></math> is defined, and the universal residual set <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub><mo>=</mo><msub><mrow><msup><mstyle mathsize='140%' displaystyle='true'><mo>&#x222A;</mo></mstyle><mtext>&#x200B;</mtext></msup></mrow><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow></msub><mi>&#x211B;</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math> is constructed. It is shown that <math><semantics><mrow><msub><mi>&#x211B;</mi><mi mathvariant='double-struck'>E</mi></msub></mrow></semantics></math> contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies <math><semantics><mrow><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x2265;</mo><mn>2</mn></mrow></semantics></math> for all <math><semantics><mrow><mi>E</mi><mo>&#x2265;</mo><mn>8</mn></mrow></semantics></math>, and that the cumulative partition count satisfies <math><semantics><mrow><munder><mstyle mathsize='140%' displaystyle='true'><mo>&#x2211;</mo></mstyle><mrow><mi>E</mi><mo>&#x2264;</mo><mi>N</mi></mrow></munder><mi>G</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&#x226B;</mo><mfrac><mrow><msup><mi>N</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mtext>log</mtext></mrow><mn>4</mn></msup><mi>N</mi></mrow></mfrac></mrow></semantics></math>. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition <math><semantics><mrow><msub><mi>E</mi><mi>n</mi></msub><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi></mrow></semantics></math> 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 <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>7</mn></mrow></semantics></math> can be expressed as <math><semantics><mrow><mi>n</mi><mo>=</mo><mi>p</mi><mo>+</mo><mn>2</mn><mi>q</mi></mrow></semantics></math>, where <math><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>&#x2208;</mo><mi>&#x2119;</mi></mrow></semantics></math>. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all <math><semantics><mrow><mi>n</mi><mo>&#x2265;</mo><mn>9</mn></mrow></semantics></math>. 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.

Abstract

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.

Abstract

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.

Abstract

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 (<b>G</b><SUB><b>p</b></SUB>) , the plaintext will be encrypted using ordinary Hill cipher technique and encrypted again using <b>G</b><SUB><b>p</b></SUB><SUB><b> </b></SUB>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.

Abstract

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.

Abstract

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to over fit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.

Abstract

The paper considers a specific type of such financial instrument as an option, namely an exotic barrier call option of the European type. Exotic options are gaining popularity among ordinary investors due to the development of information and telecommunication technologies, thanks to which such specific financial instruments as options have become readily available. We investigate the hedging problem for such options with some restrictions on the payment function and the availability of dividend payment on a risky asset in the classical Black-Scholes model. An analogue of the Black-Scholes formula for the mentioned variant of the exotic barrier is proved. In the future, it is planned to generalize the obtained results for put options and for more general payment functions.

Abstract

We study the theory of combinatorial games and find winning strategies for players. The algorithmic implementation of the winning strategies for the game TacTix is presented and the software implementation for this game in Python programming language is implemented. The program has a console interface and allows one to check the winning strategies in practice.