CAUCHY: Jurnal Matematika Murni dan Aplikasi
CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI p-ISSN: 2086-0382 | e-ISSN: 2477-3344 CAUCHY: Jurnal Matematika Murni dan Aplikasi is a scientific publication in pure and applied mathematics, including: (1) pure mathematics and mathematical structures, (2) discrete and computational mathematics, (3) applied mathematics, modeling, optimization, and control, and (4) statistics, uncertainty, and intelligent systems. This journal is published twice a year, in May and November. CAUCHY: Jurnal Matematika Murni dan Aplikasi has been accredited SINTA 2 by the Ministry of Research and Technology / National Agency for Research and Innovation of the Republic of Indonesia with Decree No. 177/E/KPT/2024, valid from May 2023 to May 2028. CAUCHY: Jurnal Matematika Murni dan Aplikasi is published by the Department of Mathematics, Faculty of Science and Technology, Universitas Islam Negeri Maulana Malik Ibrahim Malang, Indonesia, in collaboration with the Indonesian Mathematical Society (IndoMS) and Association of Pure and Education Mathematics Lecturers for Islamic Higher Education Institutions (ADMAPETA) below of Ministry Religions Affairs CAUCHY welcomes high-quality submissions of original research articles, comprehensive literature reviews, and special-topic contributions in pure and applied mathematics. All submitted manuscripts must demonstrate clear mathematical rigor, methodological soundness, and relevance to the development of pure or applied mathematics. Submitted manuscripts must explicitly identify the research gap addressed by the study and clearly state the novelty and scientific contribution of the work. The research gap must be supported by relevant scholarly literature and should explain why the study is necessary. The novelty and scientific contribution must demonstrate how the manuscript advances existing mathematical knowledge, methods, models, algorithms, or applications beyond the current state of the art. Manuscripts that do not clearly present a research gap, novelty, and scientific contribution may be considered unsuitable for review. All manuscripts must be written in English and prepared using the official CAUCHY LaTeX template available on Overleaf. Submission guidelines and detailed author instructions are available here.
Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang
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Geometry-Based Differentially Private Synthetic Tabular Data Generation via K-Means Clustering with Bounded and Discrete Feature Constraints
Most clustering-based differentially private synthetic data generation methods assume unconstrained continuous feature spaces and offer no mechanism for hard feature bound enforcement or discrete-valued attribute handling, which limits their practical applicability to real-world tabular data where such constraints are common. This paper proposes a geometry-based mechanism that generates synthetic tabular data by application of Laplace noise jointly to K-means cluster centroids and within-cluster radial distances, calibrated via a data-dependent sensitivity approximation. Three components distinguish the approach from prior work: coordinate-wise centroid reflection to enforce hard feature bounds after perturbation, coordinate-wise clipping to enforce bounds on reconstructed synthetic points, and randomized rounding for discrete features as a post-processing step. A utility-driven calibration strategy selects the privacy budget ε to meet a user-specified target Adjusted Rand Index (ARI), which makes the privacy–utility trade-off directly interpretable. Baseline comparisons on a two-dimensional illustrative example show that the proposed mechanism achieves ARI = 0.666 at ε ≈ 1.60, which substantially outperforms direct coordinate-wise noise addition at the same budget (ARI = 0.199), while it matches the non-private synthesis baseline (ARI = 0.624). Across 30 independent runs the mechanism achieves mean ARI = 0.629 ± 0.108, which confirms that the calibration target is reliably met under stochastic variation.
Copyright (c) 2026 Robby Robby https://creativecommons.org/licenses/by-sa/4.0
Negative Binomial Modeling of District-Level Diarrhea Incidence in West Java Province
This study investigated district-level diarrhea incidence in West Java Province during 2022–2024 and identified an appropriate count-data regression model for longitudinal epidemiological data. The response variable was the annual number of diarrhea cases, while explanatory variables included sanitation access, inpatient health centers, community health workers, nutritional indicators, drinking water facilities, and households implementing clean and healthy living behavior (PHBS). A population offset term was incorporated to account for differences in population exposure across districts/cities. Poisson Regression was initially fitted as a baseline model; however, severe overdispersion was detected. Therefore, Negative Binomial Regression and Generalized Linear Mixed Models (GLMM) were estimated and compared using Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), log-likelihood values, and simulation-based residual diagnostics. The results showed that Negative Binomial Regression provided the best overall fit, yielding substantially lower AIC and BIC values than competing models. Residual diagnostics indicated no evidence of remaining overdispersion, zero inflation, or serious outliers. Sanitation access was significantly associated with lower diarrhea incidence rates, with a one-standard-deviation increase corresponding to an estimated 15.7% reduction in incidence. These findings highlight the importance of accounting for overdispersion and suggest that Negative Binomial Regression is an appropriate framework for modeling regional diarrhea incidence data.
Copyright (c) 2026 Nurul Nabilah, A'yunin Sofro https://creativecommons.org/licenses/by-sa/4.0
Construction of a Family of Minimal Helicoid Surfaces Passing Through an Isoparametric Helix Curve
This research concerns the construction of a family of minimal surfaces passing through a helical curve. The construction follows the research by Yoon, which employs an approach using the curvature and torsion functions of a given curve. The construction yields a family of generalized helicoids as the minimal surfaces passing through the helical curve.
Copyright (c) 2026 Ridwan Prawira Yunaldi, Haripamyu Haripamyu, Efendi Efendi https://creativecommons.org/licenses/by-sa/4.0
Dynamical Analysis of Modified Leslie-Gower Beddington-DeAngelis Model with Prey Refuge and Intraspecific Competition
This study examines the dynamical behavior of a predator–prey model depicting the interaction between Perca fluviatilis (predator) and its natural prey, Rutilus rutilus. The model is formulated within a modified Leslie-Gower framework and employs the Beddington–DeAngelis functional response to capture mutual interference among predators during foraging. It further incorporates key ecological elements: prey refuge and nonlinear intraspecific competition arising from prey density dependence. Equilibrium points and their local stability are investigated using the Jacobian matrix and the Routh–Hurwitz criteria. The analysis identifies four equilibria: the trivial equilibrium, the predator-extinction equilibrium, the prey–extinction equilibrium, and the coexistence equilibrium. Numerical simulations corroborate these analytical results. The simulations reveal that under appropriate parameters, the stability of the system is significantly driven by both the availability of prey refuges and the degree of intraspecific competition, which ultimately determine the survival conditions for both populations.
Copyright (c) 2026 Salsabil Faridah Saputri Kusbianti, Riska Wahyu Romadhonia, Dian Savitri https://creativecommons.org/licenses/by-sa/4.0
Multivariate Time Series Forecasting of KP-RI Financial Performance Using VARIMA and VARIMAX with a Dummy Variable
Cooperative financial performance is influenced by both internal financial dynamics and external policy interventions. However, the contribution of policy-intervention information to multivariate forecasting models in cooperative finance remains insufficiently explored. This study compares Vector Autoregressive Integrated Moving Average (VARIMA) and Vector Autoregressive Integrated Moving Average with Exogenous Variables (VARIMAX-Dummy) models for analyzing and forecasting monthly Interest Income (PHB) and Net Surplus (SHU) in a savings and loan cooperative during 2015–2024. The analysis employed stationarity testing, cointegration testing, model identification, Granger causality analysis, Impulse Response Function, Forecast Error Variance Decomposition, and out-of-sample forecasting evaluation. A pulse dummy variable representing mandatory savings policy adjustments was incorporated into the VARIMAX model. The results indicate bidirectional predictive relationships and stable dynamic responses in both models. Model identification indicated that VARIMA(1,1,0) provided the optimal endogenous dynamic structure, which was subsequently retained in the VARIMAX-Dummy resulting VARIMAX (1,1,0)-Dummy specification. Forecasting evaluation showed that the VARIMAX-Dummy model achieved lower RMSE values for SHU (0.2129) and PHB (0.2088) than the corresponding VARIMA model (0.7434 and 0.6060). These findings demonstrate that incorporating policy-intervention information improves forecasting accuracy and enhances the representation of cooperative financial dynamics. The study highlights the value of exogenous policy variables in multivariate forecasting models for cooperative financial management.
Copyright (c) 2026 Tri Candra Nur Muhaimin https://creativecommons.org/licenses/by-sa/4.0
DeepSMOTE for Tea Leaf Disease Classification Using DenseNet-121 Transfer Learning
Data imbalance is a common problem in image classification that causes models to tend to be biased towards the majority class, so that the performance of the minority class decreases. To overcome this problem, this study applies the DeepSMOTE approach as a deep learning-based oversampling technique that produces synthetic images through interpolation in the latent space of the autoencoder. The balanced training data images were used to train the DenseNet-121 model through a transfer learning approach. Evaluations were carried out on three approaches, DeepSMOTE as the main approach, as well as two comparators, namely no augmentation and conventional augmentation. The results showed that DeepSMOTE had the highest accuracy, which was 97.73%, higher than the 8.99% non-augmentation approach and 4.49% from the conventional augmentation. F1-score value on minority classes such as Anthracnose increased by 0.2833 in the non-augmentation approach and 0.1318 in the conventional augmentation approach.These findings confirm that data balance plays an important role in improving model performance and that DeepSMOTE is able to provide a more even distribution of data thus supporting better generalization of image classification tasks.
Copyright (c) 2026 Nikita Putri Insani, Sugiyarto Surono, Aris Thobirin https://creativecommons.org/licenses/by-sa/4.0
Diabetes Classification Using SMOTE-Based Intuitionistic Fuzzy K-NN with Weighted HIAD
Diabetes risk classification is challenged by class imbalance and diagnostic uncertainty. This study investigates an Intuitionistic Fuzzy Set framework integrated with a proposed Hausdorff-Inspired Attribute Distance within the K-Nearest Neighbor algorithm. The framework models patient profiles using membership, non-membership, and hesitation components to capture clinical ambiguity. We evaluated this approach on the Pima Indians Diabetes dataset using Stratified 5-Fold Cross-Validation, incorporating class balancing and clinically informed feature weighting. Results showed that while a conventional Euclidean-based model achieved the highest accuracy (73.96%) and data balancing maximized sensitivity (77.61%), the intuitionistic fuzzy configurations achieved the highest specificity (89.80%), indicating a conservative uncertainty-aware classification behavior. The proposed distance measure yielded classification performance highly comparable to that of the conventional Hamming distance, providing an alternative maximum-deviation-based similarity formulation within the intuitionistic fuzzy framework. Furthermore, combining class balancing with feature weighting contributed to a more balanced sensitivity–specificity profile. Overall, the findings demonstrate how uncertainty representation, feature weighting, class balancing, and neighborhood configuration influence classification outcomes, highlighting the potential of intuitionistic fuzzy approaches for uncertainty-aware medical decision-support systems.
Copyright (c) 2026 Tio Valent Novi Yanti Sianturi https://creativecommons.org/licenses/by-sa/4.0
Robust PCA Using MCD and MM Estimators in MARS A Simulation Study
Multivariate Adaptive Regression Splines (MARS) models nonlinear relationships through adaptive basis functions but remain sensitive to outliers in the predictor variables. Existing robust extensions of MARS primarily address response outliers, while the few studies integrating Robust Principal Component Analysis (RPCA) with MARS use RPCA only for dimension reduction without comparing robust estimators. This study evaluates RPCA as a robust predictor transformation and systematically compares two robust covariance estimators—the Minimum Covariance Determinant (MCD) and the MM-estimator—within the RPCA-MARS framework. A full factorial simulation with 100 replications per condition covered 45 conditions: five sample sizes (n = 50, 100, 200, 500, 1000), three outlier proportions (5%, 10%, 25%), and three MARS interaction levels (1, 2, 3) with eight predictor variables. Outliers were extreme values in a specified proportion of predictor observations. Performance was measured by Root Mean Square Error (RMSE). For analysis, the 45 conditions were collapsed into 15 scenarios by selecting the interaction level with the minimum RMSE for each sample size and outlier proportion. The MM estimator outperformed the MCD estimator in 8 of 15 scenarios, achieving lower RMSE under moderate-to-high outlier contamination (10%–25%) with moderate sample sizes (n = 100–500). MCD performed better in the remaining 7 scenarios: under low contamination (5%) at n ≤ 200 and n ≥ 1000, and across all contamination levels at n = 1000. MCD showed higher variability at small samples with moderate-to-high contamination, while MM produced tighter confidence intervals and lower standard deviations. Within the RPCA-MARS framework, MM is recommended for moderately sized, highly contaminated data, while MCD is preferable under low contamination or in large-scale settings.
Copyright (c) 2026 Uswatun Hasanah, Solimun Solimun, Atiek Iriany https://creativecommons.org/licenses/by-sa/4.0
A Multiscale Extension of Geographically Weighted Spline Nonparametric Regression: Model Formulation and Theoretical Properties
This study proposes a Multiscale Geographically Weighted Spline Nonparametric Regression (MS-GWSNR) model to simultaneously accommodate spatial heterogeneity, multiscale spatial variation, and nonlinear relationships within a unified regression framework. The proposed model extends Geographically Weighted Spline Nonparametric Regression (GWSNR) by incorporating component-specific bandwidths adapted from the Multiscale Geographically Weighted Regression (MGWR) approach. Parameter estimation is developed using a Weighted Least Squares (WLS) framework combined with a backfitting algorithm to address the absence of a closed-form estimator under multiple spatial weighting matrices. The theoretical properties of the estimator are derived through a smoothing matrix representation, including unbiasedness, variance, and Mean Squared Error (MSE). A numerical illustration using simulated spatial data with controlled multiscale heterogeneity shows that the proposed estimator converges stably within 12 iterations, with a final smoothing operator convergence value of 2.48 × 10−5. The estimation also yields satisfactory reconstruction accuracy with MAE of 0.0680 and RMSE of 0.0880. These results indicate that the proposed MS-GWSNR model can flexibly reconstruct nonlinear spatial relationships across different spatial scales while maintaining stable estimation performance.
Copyright (c) 2026 Indi Rizqy Fahrani, Henny Pramoedyo, Atiek Iriany https://creativecommons.org/licenses/by-sa/4.0
Biclustering Performance of Iterative Signature Algorithm and Plaid Model after Imputation on Indonesian Macroeconomic Indicators
Biclustering is a two-way clustering method that identifies local patterns simultaneously across rows and columns of a data matrix. However, missing values may alter data structures and affect biclustering results. Studies evaluating the interaction between imputation methods and biclustering algorithms remain limited. This study evaluates the performance of the Iterative Signature Algorithm (ISA) and Plaid Model following imputation using Hot Deck, K-Nearest Neighbor (KNN), and Expectation Maximization (EM). The novelty of this study lies in assessing how the interaction between imputation methods and biclustering algorithms affects bicluster recovery and quality. Missing values were generated under MCAR at 5% and 10% proportions with 100 repetitions. Bicluster quality was evaluated using Mean Squared Residue (MSR), Transposed Virtual Error (VEt), and Sub-Matrix Correlation Score (SCS), while bicluster consistency was assessed using the Jaccard Index (JI). ISA consistently achieved higher JI values, indicating better preservation of bicluster structures, whereas the Plaid Model produced lower MSR, VEt, and SCS values, indicating more homogeneous biclusters. KNN generally showed the most consistent performance across scenarios. These findings suggest that imputation methods and biclustering algorithms should be selected jointly according to the analytical objective to obtain reliable biclustering results from incomplete macroeconomic data.
Copyright (c) 2026 Yani Prihantini Hiola, I Made Sumertajaya, Indahwati Indahwati https://creativecommons.org/licenses/by-sa/4.0
Optimization of Imbalanced Class Using CTGAN and RSCV-Ensemble Learning for Clean Water Quality Classification
Class imbalance is a common challenge in classification learning. This condition often leads to poor model performance in identifying minority class observations. This study aims to evaluate the performance of an Ensemble Learning model optimized using Randomized Search CV (RSCV) to classify clean water quality under imbalanced class distributions. To address the imbalance problem, the CTGAN technique is applied and compared with other oversampling methods like SMOTENC that also can handle mix data. The model performance is assessed using two sided independent t-test form different mean (accuracy, precision, recall, F1-score, and AUC-ROC) two models. The best model performance from Data Testing in this study is the Ensemble Learning model with RSCV combined with CTGAN oversampling followed by mean Accuracy of 92,8%, mean Precision of 90,23%, mean Recall of 90,98%, mean f1-score of 90,82%, and mean AUC-ROC 96,80% from 25 split data experiment. Almost all accuracy metrics showed statistically significant differences when tested using a two-sided independent t-test. Overall, the results indicate that CTGAN outperforms SMOTENC, and RSCV optimization improves performance compared to models without optimization. Not only accuracy, all different mean Recall metrics is significant, further supporting the superiority of CTGAN and RSCV-enhanced models.
Copyright (c) 2026 Favian Sis Bagus Febrianto, Umu Sa'adah, Imam Nurhadi Purwanto https://creativecommons.org/licenses/by-sa/4.0
Existence and Uniqueness Property On a Generalized Ledin-Brousseau Sum
In this paper, we present the existence and uniqueness property of the Ledin-Brousseau sum involving a polynomial and a homogeneous linear recurrence sequence. This sum is of the form ∑k=1n P(k)shk+r, where n is a positive integer, P(x) is a polynomial in C[x], h and r are integers, and (sk)k∈Z is a homogeneous linear recurrence sequence of degree m ≥ 2 satisfying certain constraints.
Copyright (c) 2026 Ivan Hadinata, Muhamad Abdillah Ahen https://creativecommons.org/licenses/by-sa/4.0
Danantara Effects on IDXESGL Stock Investment Strategies Using Single Index Model and VaR-Adjusted Sharpe Ratio
Danantara, established as Indonesia’s sovereign wealth fund on February 24, 2025, has introduced new dynamics in the Indonesian capital market. This study examines differences in portfolio strategies among IDX ESG Leaders (IDXESGL) index between the pre- and post-Danantara periods by comparing stock composition with positive expected returns and portfolio performance across two periods using the Single Index Model (SIM) and the Value at Risk-Adjusted Sharpe Ratio (VaRSR). The analysis employs an integrated framework that combines SIM for identifying best-performing stocks, a portfolio combination approach, and performance evaluation using the Sharpe ratio and VaRSR. This study utilized daily closing price data covering August 23, 2024, to February 23, 2025, as the pre-Danantara period, and February 24, 2025, to August 24, 2025, as the post-Danantara period. The results indicate that prior to Danantara’s establishment, optimal portfolios were dominated by digital-sector stocks, with higher performance than in the post-Danantara period. In contrast, the post-Danantara period is associated with greater exposure to conglomerates and state-owned enterprises with stronger fundamentals and lower downside risk. Overall, the findings suggest that the VaRSR yields a more conservative performance assessment and highlights a potential reorientation in investment strategies toward stability and resilience following Danantara’s establishment.
Copyright (c) 2026 Risky Gunawan, Evy Sulistianingsih, Shantika Martha, Yundari Yundari, Fida Fathiyah Addini https://creativecommons.org/licenses/by-sa/4.0
Coefficient Bounds for a Subclass of Bi-Univalent Functions
In this paper, we obtain explicit upper bounds for the initial seven coefficients and their inverses for a subclass of bi-univalent functions defined via subordination. Using coefficient estimates associated with the Carathéodory class and coefficient comparison techniques, expressions for the bounds of a2, a3, …, a7 are derived. Moreover, it is proved that the first seven inverse coefficients admit the same upper bounds as the corresponding coefficients of the original function. These findings generalize several existing results concerning lower-order coefficients and provide a framework for further investigations of higher-order coefficient and Hankel determinant problems in the theory of bi-univalent functions.
Copyright (c) 2026 Marjono Marjono https://creativecommons.org/licenses/by-sa/4.0
Negative Binomial tv-DINGARCH Models for Monthly DHF Cases in Surabaya, Indonesia (2015-2025)
Count time series data are commonly encountered in epidemiological studies, such as monthly reports of Dengue Hemorrhagic Fever (DHF) cases. The INGARCH model has been widely used to handle temporal dependence and overdispersion in such data. However, the standard INGARCH model assumes constant dispersion, whereas in practice the dispersion level may vary over time due to epidemiological factors like seasonality or vector control policies. Accurately capturing this time-varying variability is essential for reliable disease surveillance. This study applies the tv-DINGARCH model to DHF cases in Surabaya from January 2015 to December 2025. The tv-DINGARCH(1,1,1,1) model with Negative Binomial distribution is used because it allows the dispersion parameter \(\phi_t\) to evolve dynamically. Parameter estimation is conducted using the Conditional Maximum Likelihood Estimation method implemented in RStudio. The results show that the tv-DINGARCH model yields AIC = 968.43 and BIC = 985.68, both smaller than those of the standard INGARCH model (AIC = 1012.18, BIC = 1023.68). The likelihood ratio test produces an LR statistic of 47.7560 with a p-value less than 0.000001, strongly rejecting constant dispersion. The stationarity condition is satisfied with \(\beta_2 + \alpha_2 + \beta_1 + \alpha_1 = 0.9990 < 1\). From a public health perspective, these findings indicate that the tv-DINGARCH model better captures the fluctuating dynamics of DHF transmission, supporting more effective surveillance and early warning systems.
Copyright (c) 2026 Aminatuz Zahro, A'yunin Sofro https://creativecommons.org/licenses/by-sa/4.0
Spatial Heterogeneity of Poverty Determinants in Indonesia A Hierarchical Geographically Weighted Regression Approach
This study employs the Hierarchical Geographically Weighted Regression (HGWR) model to analyze poverty determinants in Indonesia, addressing spatial heterogeneity and hierarchical data structures simultaneously. Using data across 34 provinces and 508 regencies/cities, the HGWR model with a Gaussian kernel (bandwidth = 15) substantially outperforms the global Ordinary Least Squares (OLS) regression, increasing the R2 value from 0.502 to 0.754. The Moran's I test on the global model residuals (0.2216, p < 0.001) justifies the urgency of accounting for spatial nonstationarity, while the HGWR post-estimation residuals show that spatial autocorrelation is successfully eliminated (-0.000641, p = 0.416). At the regency/city level, adjusted per capita expenditure and the poverty line significantly reduce the poverty headcount rate, whereas the average years of schooling shows no significant localized effect. At the provincial level, the Human Development Index (HDI) consistently reduces poverty (mean coefficient of -0.8082) but exhibits substantial spatial variation, where the impact is strongest in eastern Indonesia (coefficients < -0.95) and weakest in Java (coefficients > -0.65). Conversely, expected years of schooling exhibits a positive mean coefficient (3.4755), with its positive effects highly concentrated in Java. These findings conclude that poverty reduction strategies in Indonesia must be place-based rather than uniform, prioritizing provincial HDI improvements in eastern Indonesia where the marginal returns of development policy are highest.
Copyright (c) 2026 Debora Dwi Kurniawati, Henny Pramoedyo, Suci Astutik https://creativecommons.org/licenses/by-sa/4.0
Bayesian Framework for Error Correction Model-Nonlinear Autoregressive Distributed Lag
This study proposes a Bayesian framework for estimating the Error Correction Model–Nonlinear Autoregressive Distributed Lag (ECM–NARDL) to analyze the dynamic and asymmetric relationship between inflation and economic growth in Indonesia. The model is estimated using Gibbs Sampling within a Markov Chain Monte Carlo (MCMC) framework, allowing parameter uncertainty to be evaluated through posterior distributions. Annual data from 1990–2024 are used for empirical analysis. Unit root tests indicate that the variables are integrated of order one, while the Bounds test confirms the existence of a long-run equilibrium relationship. The estimation results reveal a significant error-correction mechanism, suggesting that deviations from long-run equilibrium are corrected relatively quickly. The findings also indicate asymmetric effects of inflation, where decreases in inflation support long-run economic growth, while short-run increases in inflation negatively affect growth. Bayesian diagnostics and posterior predictive checks confirm the stability and adequacy of the proposed Bayesian ECM–NARDL framework.
Copyright (c) 2026 Meidy Indhira Putri, Nurjannah Nurjannah, Achmad Efendi, Ani Budi Astuti https://creativecommons.org/licenses/by-sa/4.0
Hybrid ARIMAX–LSTM Modeling with Elastic Net for Time Series Forecasting the Education Index of Jambi Province
Abstract This study proposes a Hybrid ARIMAX–LSTM framework integrated with Elastic Net regularization to improve the forecasting accuracy of the Education Index of Jambi Province. Annual data from 2010 to 2024 are utilized, with Elastic Net applied to select relevant exogenous variables and mitigate multicollinearity. The modelling strategy combines an ARIMAX (1,0,0) model to capture linear dynamics and the influence of selected explanatory variables, with a Long Short-Term Memory (LSTM) network to learn nonlinear patterns from ARIMAX residuals. The dataset is divided into training and testing subsets, where the model is trained using historical data and subsequently validated on unseen observations to assess its generalization capability. Forecasting performance is evaluated using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE) on out-of-sample data for 2023 2024. Empirical results show that the proposed hybrid model with Elastic Net significantly outperforms the hybrid model without regularization, achieving an RMSE of 0.3677, an MAE of 0.2647, and a MAPE of 0.40%. In contrast, the ARIMAX–LSTM model with Elastic Net demonstrates substantially better predictive performance, with an RMSE of 0.0720, an MAE of 0.0714, and a MAPE of 0.11%. These findings demonstrate that Elastic Net enhances model generalization and improves residual learning in the LSTM component. After model evaluation, the hybrid models are further employed to generate forecasts of the Education Index for the period 2025–2027 to support medium-term educational planning. Overall, the proposed framework provides a robust and interpretable approach for forecasting regional educational indicators.
Copyright (c) 2026 Mhd. Teguh Arrozik https://creativecommons.org/licenses/by-sa/4.0
Dynamic Analysis of a Modified SETPR Terrorism Model Incorporating Recidivism of Imprisoned Terrorists
Terrorism poses a serious threat to national security, particularly when former terrorist inmates return to active duty due to the failure of deradicalization programs. This study modifies the $SETPR$ model by adding a direct recidivism path from the imprisoned compartment (\(P\)) to the active terrorist compartment (\(T\)) via the parameter \(k\). This model consists of five subpopulations: the susceptible (\(S\)), the exposed (\(E\)), active terrorists (\(T\)), imprisoned (\(P\)), and subpopulation that has ceased terrorist activities (\(R\)). The basic reproduction number is given by \(\mathcal{R}_0,\) which is derived using the next-generation matrix method and serves as a threshold parameter for the spread of terrorism. Local stability analysis shows that the terrorism-free equilibrium is locally asymptotically stable if \(\mathcal{R}_0 < 1\), while the endemic equilibrium is locally asymptotically stable if \(\mathcal{R}_0 > 1\). Numerical simulations using Matlab R2013a and Maple 11 show that when \(\mathcal{R}_0 = 0.0039 < 1\), the system converges to the terrorism-free equilibrium, meaning terrorism will become extinct. Conversely, with \(\mathcal{R}_0 = 5.8535 > 1\), the system is stable at the endemic equilibrium, indicating that terrorism persists at a positive level. These results show that it is important to control key parameters through integrated intervention strategies, so that the \(\mathcal{R}_0\) value can be reduced below unity.
Copyright (c) 2026 Achla Fauziyah, Zulaikha Zulaikha https://creativecommons.org/licenses/by-sa/4.0
A Multiplicative Extension of Weighted Hadamard Fractional Integrals
This paper presents an innovative category of weighted Hadamard fractional integral operators situated within the paradigm of multiplicative calculus. The operator introduced herein broadens the traditional weighted Hadamard fractional integral by integrating the multiplicative framework of non-Newtonian calculus, thereby establishing a cohesive linkage between weighted fractional analysis and multiplicative fractional calculus. We delineate the formulation of both the left- and right-sided multiplicative weighted Hadamard fractional integral operators and explore their core analytical characteristics. Specifically, a multiplicative linearity property is demonstrated, the continuity of the logarithmic representation of the operator is affirmed, and boundedness results are extracted in weighted Lebesgue-type spaces through the application of Hölder's inequality. Furthermore, we establish that the proposed operators comply with a semigroup property, which ensures alignment with the classical framework of fractional integration. These findings illustrate that the introduced operators maintain crucial structural attributes of weighted fractional integrals while seamlessly extending them to the multiplicative context. The theoretical framework developed herein lays a foundational basis for subsequent explorations in multiplicative fractional calculus, fractional integral inequalities, and associated applications within the realm of mathematical analysis.
Copyright (c) 2026 Almeer Naufal Hakim, Marjono Marjono, Corina Karim https://creativecommons.org/licenses/by-sa/4.0
Adaptive Portfolio Rebalancing for NASDAQ-100 Stocks with Hippopotamus Optimization Algorithm under Transaction Costs
This study proposes a transaction cost-aware portfolio optimization framework for NASDAQ-100 stocks based on the Hippopotamus Optimization Algorithm (HOA). The mathematical model extends the classical mean-variance framework by incorporating transaction costs into a Net Sharpe Ratio (NSR) objective function. Daily adjusted closing prices of NASDAQ-100 constituent stocks from 2021 to 2025 are employed, with the twenty highest-ranked stocks according to the Sharpe Ratio (SR) selected as the candidate investment universe. Each optimization experiment is independently repeated 25 times to evaluate robustness and solution stability. The results indicate that the proposed HOA consistently achieves competitive optimization performance with very small variability across repeated runs throughout the investment horizon. Although several competing algorithms attain comparable or slightly better objective values during particular rebalancing periods, HOA remains among the strongest-performing methods while exhibiting stable convergence behavior. Throughout the investment horizon, the proposed algorithm produces the highest average cumulative portfolio value of \$5,307.87 ± \$125.71 from an initial investment of \$1,000, corresponding to an average annual return of 40.03% ± 0.68%, together with the highest average SR and NSR of 1.5736 ± 0.0151, while maintaining competitive maximum drawdown and transaction costs. These findings demonstrate that incorporating transaction costs directly into the optimization objective enables HOA to achieve a favorable balance between portfolio growth, risk-adjusted performance, and trading efficiency under dynamic market conditions.
Copyright (c) 2026 Mohamad Muslikh https://creativecommons.org/licenses/by-sa/4.0
Macroeconomic Determinants of Investment Credit in Indonesia: Evidence from the ARDL Bounds Testing Approach
Investment credit plays an important role in financing productive activities and sustaining Indonesia's economic development. Nevertheless, limited empirical evidence is available regarding how fluctuations in gold prices together with other macroeconomic indicators influence investment credit during the post-pandemic period. This study investigates the effects of gold prices, the USD/IDR exchange rate, the Industrial Production Index (IPI), the BI 7-Day Reverse Repo Rate, and inflation on investment credit using monthly observations from June 2016 to December 2024. An Autoregressive Distributed Lag (ARDL) model combined with an Error Correction Model (ECM) is employed to evaluate both long-run associations and short-run adjustments. The empirical findings reveal that the variables are cointegrated, implying the existence of a stable long-term equilibrium. However, none of the estimated long-run coefficients is statistically distinguishable from zero at conventional significance levels. In the short run, exchange rate movements generate the largest response in investment credit, whereas industrial production and the policy interest rate produce relatively modest effects. The error-correction coefficient is negative and statistically significant, indicating that temporary departures from equilibrium are gradually eliminated over time. These findings suggest that investment credit in Indonesia is driven primarily by short-term macroeconomic adjustments rather than persistent long-run effects of individual macroeconomic variables.
Copyright (c) 2026 Muhammad Ilham Arif, Nur Atikah, Muhammad Nizar Hidayatullah, Mareta Putri Manunggal, Vagustin Faharani https://creativecommons.org/licenses/by-sa/4.0
A Generalization Euclidean Algorithm Approach for Solving Multi-Variable Linear Diophantine Equations
Multi-variable linear Diophantine equations of the form a1x1 + a2x2 + ··· + anxn = b, where ai, b ∈ Z, have various applications in many fields, including cryptography, chemistry, and statistics, where they can be used to determine public and private keys, balance chemical equations, and model scheduling problems, respectively. The equation has infinitely many integer solutions if gcd(a1, a2, …, an) divides b. Two well-known algorithms for finding solutions are the Smith normal form and integer lattice methods. This paper presents an alternative approach for obtaining the general solution of the equation. The proposed method applies the generalized Euclidean algorithm to compute gcd(a1, a2, …, an), followed by a back-substitution process through the algorithm's steps to express the greatest common divisor as a linear combination of a1, a2, …, an. This linear combination is then multiplied by b/gcd(a1, a2, …, an) to obtain a particular solution of the equation. Finally, the general solution is constructed from the resulting particular solution.
Copyright (c) 2026 Ikhsan Abdul Ro'uf, Hery Susanto, I Made Sulandra https://creativecommons.org/licenses/by-sa/4.0