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.

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.

In this paper we present a new classes of solutions for the Einstein-Maxwell system of field equations in a spherically symmetric spacetime under the influence of an electric field considering the MIT bag model equation of state with a particular form the metric potential that depends on an adjustable parameter. The obtained solutions can be written in terms of elementary functions, namely polynomials and algebraic functions. The obtained models satisfy all physical properties expected in a realistic star. The results of this research can be useful in the development and description of new models of compact structures.

The future of e-money is crypocurrencies, it is the decentralize digital and virtual currency that is secured by cryptography. It has become increasingly popular in recent years attracting the attention of the individual, investor, media, academia and governments worldwide. This study aims to model and forecast the volatilities and returns of three top cryptocurrencies, namely; Bitcoin, Ethereum and Binance Coin. The data utilized in the study was extracted from the higher market capitalization at 31^st December, 2021 and the data for the period starting from 9^th November, 2017 to 31^st December 2021. The Generalised Autoregressive conditional heteroscedasticity (GARCH) type models with several distributions were fitted to the three cryptocurrencies dataset with their performances assessed using some model criterion tests. The result shows that the mean of all the returns are positive indicating the fact that the price of this three crptocurrencies increase throughout the period of study. The ARCH-LM test shows that there is no ARCH effect in volatility of Bitcoin and Ethereum but present in Binance Coin. The GARCH model was fitted on Binance Coin, the AIC and log L shows that the CGARCH is the best model for Binance Coin. Automatic forecasting was perform based on the selected ARIMA (2,0,1), ARIMA (0,1,2) and the random walk model which has the lowest AIC for ETH-USD, BNB-USD and BTC-USD respectively. This finding could aid investors in determining a cryptocurrency's unique risk-reward characteristics. The study contributes to a better deployment of investor’s resources and prediction of the future prices the three cryptocurrencies.