https://gphjournal.org/index.php/m <p style="font-family: 'Segoe UI', sans-serif; font-size: 16px; color: #333;"><strong>GPH-International Journal of Mathematics (e-ISSN&nbsp;<a href="https://portal.issn.org/resource/ISSN/3050-9629" target="_blank" rel="noopener">3050-9645</a>)</strong> is a peer-reviewed, open-access journal dedicated to advancing research in mathematics. The journal publishes original research articles, comprehensive reviews, and survey papers covering both pure and applied mathematics, including topics such as algebra, analysis, geometry, number theory, and mathematical modeling. It provides a global platform for mathematicians and researchers to share innovative ideas, foster interdisciplinary collaboration, and contribute to the advancement of mathematical knowledge.</p> Global Publication House en-US GPH-International Journal of Mathematics 3050-9645 <p>The authors and co-authors warrant that the article is their original work, does not infringe any copyright, and has not been published elsewhere. By submitting the article to <a class="is_text" title="GPH - International Journal of Mathematics" href="https://www.gphjournal.org/index.php/m/index">GPH - International Journal of Mathematics</a>, the authors agree that the journal has the right to retract or remove the article in case of proven ethical misconduct.</p> https://gphjournal.org/index.php/m/article/view/2614 <p>This study models the volatility of the global Brent crude oil price and examines its implications for the Nigerian economy, which depends heavily on crude oil exports for fiscal revenue and foreign exchange earnings. Using a monthly Brent crude oil spot price series (Europe) obtained from the U.S. Energy Information Administration via the Federal Reserve Economic Data (FRED) database, spanning May 1987 to July 2026 (471 price observations; 470 monthly log-returns), the study compares three GARCH-family conditional-variance models of increasing flexibility – symmetric GARCH(1,1), EGARCH(1,1) and GJR-GARCH(1,1), each with Student-t innovations – against a single-layer Long Short-Term Memory (LSTM) neural network built and validated from first principles. Descriptive statistics show pronounced excess kurtosis (9.06) and negative skewness (−0.49), with the Jarque-Bera statistic (719.84, p&lt;.001) decisively rejecting normality. The Brent price level is non-stationary (ADF t = −1.72) while its return series is stationary (ADF t = −9.02), and both the Ljung-Box test on squared returns (Q = 210.00, p&lt;.001) and the ARCH-LM test (LM = 171.78, p&lt;.001) confirm strong volatility clustering. Formal structural-break tests at two independently documented oil-market events – the November 2014 OPEC non-cut decision and the March 2020 COVID-19 demand collapse – each find a highly significant post-break increase in return variance (F = 2.034 and F = 2.552, both p&lt;.001) but no significant shift in the mean return (p = .616 and p = .839). Fitted on an 80% training window (n = 376), EGARCH(1,1) achieves the best in-sample fit by AIC (2653.32) while GARCH(1,1) achieves the best BIC (2675.44); neither asymmetric model's leverage coefficient is statistically significant (EGARCH γ = −0.063, t = −1.16; GJR-GARCH γ = 0.111, t = 0.98), although EGARCH's negative sign is stable across an independent post-2015 robustness re-fit. On a 94-month out-of-sample test window, EGARCH(1,1) achieves the lowest RMSE (427.77) and MAE (149.88), GJR-GARCH(1,1) the lowest Theil's U (0.529), and the LSTM the lowest MAPE (272.89%); a Diebold-Mariano test comparing EGARCH against the LSTM is not statistically significant (DM = −1.067, p = .286). The study concludes that Brent crude oil price volatility is highly persistent and asymmetric in sign, though not significantly so, and that a well-specified asymmetric GARCH model remains competitive with, and is not clearly outperformed by, a deep-learning alternative, a finding of direct relevance to Nigerian fiscal and reserve-management planning.</p> Mohammed Anono Zubair James Oyaonu Esther ##submission.copyrightStatement## https://creativecommons.org/licenses/by-nc-nd/4.0 2026-10-13 2026-10-13 9 9 01 23 10.5281/zenodo.23339193 https://gphjournal.org/index.php/m/article/view/2625 <p>The UHI (Urban Heat Island) problem in Karbala, Iraq is complicated and extends beyond normal city maps. To better understand this problem, the researchers utilized QRS (Quantitative Remote Sensing) along with TDA (Topological Data Analysis), treating the thermal field(s) like a Topological Manifold. The researchers used Vietoris-Rips filtering on Landsat-8 imagery to make "Thermal Shape Fingerprints" of the UHI (Urban Heat Island); at a global average temperature of 48.7°C (but many structured thermal features were found), Persistent Homology was calculated, and it became evident that a transition was located at a thermal level of 42°C, giving rise to 7 main thermal loops (beta1 generation elements) encircling the centre.</p> <p>The researchers also found that the stable H1 cycle situated at the shrines proved that the architectural elements of these structures acted as a "topological void." This effect produces (~14°C (Delta QA)) in cooling from the surrounding ambient temperature. This effect was also supported by the calculation of Wasserstein Stability of 3.1, confirming that the structures themselves are fixed and presented the requirement for "Topological Urban Design" to separate the 18 stable thermal clusters.</p> Hameed Hasan Obaid Anous Yasari ##submission.copyrightStatement## https://creativecommons.org/licenses/by-nc-nd/4.0 2026-10-13 2026-10-13 9 9 24 42 10.5281/zenodo.23340970