Cauchy

This study examines the spatial spillover dynamics of the Human Development Index (HDI) in West Java Province, Indonesia, using the Enclave Panel Threshold Spatial Autoregressive (E-PT-SAR) model and panel data from 27 districts during 2021–2025. The model extends the conventional PT-SAR framework by incorporating enclave identification, distinguishing core prosperous regions, prosperous enclaves, and poor regions. The estimated poverty threshold (γ = 8.5124%) separates West Java into prosperous and 13 poor districts, while the enclavity index and a stationarity-constrained enclave threshold (τ = 0.7143) identify a single prosperous enclave, Ciamis. The results reveal that spatial spillovers differ markedly across regimes. Crucially, the spillover among prosperous districts is insignificant in the conventional two-regime model (ρ = 0.4656, p = 0.240) but becomes significant once the isolated enclave is separated (core prosperous ρ = 0.6344, p = 0.025), revealing a spillover that the standard threshold model masks. A strong spillover is also found in the poor regime (ρ = 0.9740), whereas the enclave shows no statistically detectable spillover, consistent with the hypothesis that spatially isolated prosperity does not diffuse to surrounding poorer areas. The effects decomposition confirms that indirect (spillover) effects dominate in the poor regime. Because the enclave regime comprises only one district, its results are interpreted as exploratory. These findings imply that regional development policies should be tailored to different spatial regimes. Keywords: Spatial Econometrics; E-PT-SAR; Enclave; Spatial Spillover; HDI; West Java

Understanding relationships among products in retail transactions is essential for supporting strategic decisions such as product placement and cross-selling. However, traditional market basket analysis is limited to pairwise associations and cannot capture complex conditional dependencies among multiple items. This study addresses these limitations by applying a Bayesian Network framework to model product relationships probabilistically. Transaction data collected over one month were transformed into a binary dataset of 441 transactions across 31 product categories. The network structure was learned using a stochastic search algorithm with a Bayesian Dirichlet equivalent (BDe) score, enhanced by a multi-seed strategy and edge support aggregation to ensure robustness. Parameters were estimated using maximum likelihood. The results indicate that relationships are both directional and context-dependent, with snack probabilities increasing with instant noodles but decreasing when cigarettes are also present. The LOIO accuracy is 0.6659.

Pollutant dispersion in rivers is governed by advection, diffusion, and the physical characteristics of the channel. This paper models two-dimensional pollutant transport using the advection-diffusion equation and solves it numerically with the Finite Element Method (FEM) under five scenarios: constant flow with a single pollutant source, flow that follows a meandering channel, constant flow with two sources, the presence of a rock obstacle, and an irregular river domain. Simulations are implemented in Mathematica through domain construction, mesh generation, and a Finite Element-based numerical solution. The results show that flow velocity is the primary driver of plume movement, while diffusion smooths concentration gradients. Comparative analysis across the five scenarios demonstrates that obstacle-containing and irregular domains produce the widest plume spreading and the strongest concentration deformation compared to the straight-channel case. Peak concentrations also decrease more rapidly in multi-source and irregular-flow scenarios due to enhanced mixing and plume interaction. Physical obstacles and channel irregularities generate loacal recirculation zones and plume deviation, producing more realistic pollutant transport behavior than simplified channer models. These findings highlight the importance of geometry-aware flow representations for understanding river pollutant transport in numerical modelling studies.

Online shopping addiction is a growing behavioral problem among university students, driven by the rapid development of e-commerce platforms. This study constructs, analyzes, and applies optimal control to an SEIRS (susceptible--exposed--infected--recovered--susceptible) mathematical model of online shopping addiction dynamics among students of the Faculty of Mathematics and Natural Sciences, Universitas Negeri Makassar. The population is divided into susceptible, exposed, addicted, and recovered compartments, and the model incorporates the possibility of relapse from the recovered to the susceptible compartment. The analysis determines the addiction-free and endemic equilibria, examines local stability through Jacobian linearization, and computes the basic reproduction number using the next-generation matrix method. Optimal control is formulated using Pontryagin's minimum principle, with an educational and counseling intervention as the control applied to the addicted compartment. Primary data were obtained from questionnaires distributed to ninety-eight active students, and numerical simulations were carried out using the fourth-order Runge--Kutta and forward--backward sweep methods in Python. The basic reproduction number is approximately 1.03, which is greater than one, indicating that online shopping addiction can persist and spread within the population; consistently, the addiction-free equilibrium is unstable and the endemic equilibrium is locally asymptotically stable. Applying the optimal control suppresses the peak addicted population by 43.1 percent, accelerates the movement of addicted individuals toward recovery during the intervention window, and preserves a substantially larger never-addicted susceptible population. These results demonstrate that educational intervention is mathematically effective in controlling the spread of online shopping addiction.

Forest resources are essential for maintaining ecological balance and supporting human livelihoods, but population growth and mining expansion have intensified forest degradation. This study develops a nonlinear dynamic model of forest resource management involving forest density, human population, population pressure, and mining activities, and determines optimal intervention strategies for sustainable management. The model is formulated as a four-dimensional system of nonlinear ordinary differential equations with two control variables: environmental education and mining restrictions. The existence of optimal controls is established using the Filippov–Cesari theorem, while Pontryagin’s Maximum Principle is applied to characterize the optimal solutions. Numerical simulations using the forward–backward sweep method show that environmental education reduces population pressure by 9.57%, whereas mining restrictions reduce mining activities by 81.23% and increase forest density by 70.41%. Simultaneous implementation of both controls provides the best outcome, improving forest sustainability while balancing ecological conservation and socioeconomic development objectives.

This study presents a sequential hybrid spatio-temporal forecasting framework combining the Generalized Space-Time Autoregressive with Exogenous Variables (GSTARX-IDW) model and Ordinary Kriging (OK) to model and map weekly Fire Radiative Power (FRP) dynamics in West Kalimantan from January 2021 to October 2025. A strong dominance of spatial contagion was observed, with the spatial autoregressive parameter (p1) being statistically significant across 95.56% of operational grid centroids, providing empirical validation of Tobler's First Law of Geography. Locally, Land Surface Temperature (LST) serves as a key exogenous forcing variable, exhibiting geographical dichotomies driven by localized microclimatic conditions and peatland hydrology. To overcome the limitation of discrete point forecasts at grid centroids, Ordinary Kriging was applied directly to the k-step ahead GSTARX-IDW point forecasts, successfully reconstructing continuous spatial risk surfaces for October 2025. Evaluated through robust out-of-sample metrics, the framework achieved a Root Mean Squared Error (RMSE) of 1.1380, a Mean Absolute Error (MAE) of 0.8736, and a Mean Absolute Scaled Error (MASE) of 1.0008, demonstrating competitive temporal point forecasting on par with baseline dynamics while offering superior spatial continuous risk mapping. This sequential framework provides a mathematically grounded baseline for short-term spatio-temporal risk assessment in highly fragmented tropical landscapes.

A three-compartment predator-prey model (prey, middle predator, top predator) is developed by integrating spatial diffusion, discrete time delays (gestation and biomass conversion), and proportional harvesting on the two lower trophic levels. The objectives are to formulate the model, analyze local stability dynamics and Hopf bifurcation due to time delay variations, and determine a sustainable harvesting strategy based on the Maximum Sustainable Yield (MSY) concept. Methods include equilibrium analysis, linearization using a variation matrix, Routh–Hurwitz criterion, and numerical simulations with GNU Octave in one- and two-dimensional spatial domains. The results show that a positive interior coexistence equilibrium exists and is locally asymptotically stable under certain conditions. Single or double time delays without harvesting trigger Hopf bifurcation when exceeding critical values, while pure diffusion does not produce instability on its own. Harvesting at safe rates increases the critical delay values and maintains coexistence, in contrast to the mathematical MSY harvesting that leads to species extinction. In conclusion, the integration of diffusion, time delays, and harvesting yields complex dynamics, with time delays as the main destabilizer and controlled harvesting as a stabilizing agent that delays the onset of Hopf bifurcation.

This paper proposed a new model to price a stock option based on the Skewed Laplace distribution approach (SLOP). The approach was considered to provide a better option price than Black Scholes Option Price (BSOP) because Skewed Laplace distribution (SL) has a shape parameter that can capture excess skewness and kurtosis frequently found in stock return underlying the option price. In this study, SL’s shape parameter was estimated using a mixture of the Moment Method and Fourth-Order Taylor Series approach. The estimator was different from the majority of prior SL’s shape parameter that was obtained by Maximum Likelihood Estimation (MLE). The proposed shape parameter was easier to obtain relative to the prior parameter because it did not require a Likelihood Function (LH) and the maximization of LH where involved a complicated numerical method. The performance of SLOP was applied to eleven different enterprises that trade stock options at several strike prices. According to the empirical results in this research, it can be summarized that the SL approach yields a better option price model rather than Black Scholes (BS).

This study investigates the local dynamics of a discrete-time host–parasitoid model incorporating a proportional host refuge mechanism. The objective is to clarify how host growth and refuge intensity govern the transitions among extinction, stable coexistence, and oscillatorydynamics. Linearization, stability conditions, discriminant analysis, and local normal-form1calculations are used to determine the existence and stability of the extinction and coexistencefixed points and to characterize their local behavior. The boundary between extinction andcoexistence is identified as a nonhyperbolic threshold supporting a continuum of boundaryfixed points. Within the stable coexistence region, the discriminant separates node-typeconvergence from damped spiral convergence. Variation in the refuge fraction is shown toinduce a supercritical Neimark–Sacker bifurcation, producing a stable invariant closed curveand bounded oscillations on the weak-refuge side near the critical threshold. Numerical phaseportraits, time series, bifurcation diagrams, and a two-parameter map support the analyticalfindings. These results provide a unified description of extinction, stable coexistence, andoscillatory behavior in the refuge model.

The classical Black-Scholes model assumes that stock prices follow a Geometric Brownian Motion (GBM) process, in which stock prices remain strictly positive throughout time. However, this assumption restricts the model’s capability to represent firms experiencing financial distress, where stock prices may gradually approach zero as bankruptcy risk increases. This study employs the Extended Black-Scholes Model (EBSM), which introduces a stoppingtime mechanism that enables stock prices to reach the bankruptcy boundary, to analyze and compare bankruptcy risk characteristics between financially healthy and distressed stocks. Historical adjusted closing price data from Microsoft Corporation (MSFT.US) and Bed Bath Beyond Inc. (BBBY.US) were utilized, with model parameters estimated using quadratic variation and Maximum Likelihood Estimation (MLE) methods. The estimation results revealed different stochastic characteristics between the two stocks, where MSFT.US generated a positive drift parameter of 0.216682, whereas BBBY.US generated a negative drift parameter of −0.248385 along with higher volatility. The bankruptcy risk analysis demonstrated that MSFT.US produced infinite Expected Bankruptcy Time (EBT) and Conditional Expected Bankruptcy Time (CEBT) values, while BBBY.US yielded a finite EBT of 5.113633 years, with CEBT values updated according to the observed stock price conditions. These results suggest that the EBSM framework can effectively differentiate bankruptcy risk characteristics by combining stochastic stock price dynamics with time-to-bankruptcy measures.

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.

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.

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.

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.

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.

Measles remains a significant public health challenge in Indonesia, especially in regionswith heterogeneous vaccination coverage and waning immunity. This study develops andanalyzes a Susceptible–Vaccinated–Infected–Recovered (SVIR) compartmental model formeasles transmission in Sidoarjo City, Indonesia, incorporating waning immunity. The basicreproduction number R0 is derived using the next-generation matrix method, and the globalstability of both the disease-free and endemic equilibria is rigorously proven via Lyapunovfunctions. Model parameters are estimated from weekly 2023 surveillance data in Sidoarjo Cityusing Particle Swarm Optimization (PSO), Extended Kalman Filter, Maximum LikelihoodEstimation, and Least Squares, and evaluated via Mean Absolute Percentage Error (MAPE).PSO achieves the lowest MAPE (11.17%), outperforming the other methods. The estimatedR0 = 0.668confirms a disease-free regime. Normalized sensitivity analysis identifies vaccineefficacy as the most influential parameter; a decline in efficacy to0 .75elevates R0 close tothe critical threshold, and any further reduction pushes the system into an endemic state.These findings underscore the necessity of maintaining high vaccine efficacy through boosterprograms and strengthened immunization strategies to sustain measles elimination at thedistrict level.

Volatility modeling plays a crucial role in time-series forecasting, particularly for wind speed, where variability and asymmetric responses to shocks are commonly observed. Accurate wind speed forecasting can help mitigate potential risks associated with extreme or uncontrolled wind events. While the Autoregressive Integrated Moving Average (ARIMA) model is widely used to model the conditional mean of time series, it does not capture time-varying volatility or asymmetric effects. To address this limitation, we combine ARIMA with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and asymmetric extensions, including the Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) and its generalized form (GGJR-GARCH). This framework allows simultaneous modeling of the conditional mean and conditional variance, accommodating heteroscedasticity and leverage effects in wind speed data. The empirical results indicate that negative shocks exert a stronger impact on conditional volatility than positive shocks, confirming the presence of asymmetry. Based on forecasting performance evaluation, the ARIMA(2,0,1)–GGJR-GARCH(1,1) specification provides the most accurate predictions among the competing models.

This study develops a three-dimensional mangrove detritus small fish model with a dynamics harvesting effort governed by a threshold rule and a Beddington DeAngelis functional response. The main objective is to understand how adaptive harvesting and consumer interference shape long-term dynamics and stability. Equilibrium points are derived and their local stability is examined using the Jacobian matrix and eigenvalue-based criteria, supported by numerical simulations. Bifurcation analysis with respect to the threshold parameter is performed using numerical continuation to detect qualitative transitions in system behavior. The results show the existence of biologically feasible coexistence equilibria with active harvesting, including a stable branch and an unstable branch. A Hopf bifurcation is detected at approximately a = 6.6159, confirming the emergence of sustained oscillations limit cycles in fish density and harvesting effort, while a fold limit point occurs near a = 22.001, indicating changes in equilibrium structure. Moreover, simulation scenarios demonstrate that reducing environmental capacity and increasing fish mortality can drive the system toward a no-effort equilibrium where harvesting collapses. These findings highlight the threshold parameter as a key control factor that can switch the system between stable harvesting, effort extinction, and oscillatory harvesting regimes.

High-frequency wind speed data collected via environmental monitoring systems often contain significant stochastic noise that can obscure underlying patterns and degrade the reliability of statistical models. A two-stage modeling framework (integrating a Kalman Filter (KF) for signal purification and Autoregressive Integrated Moving Average (ARIMA) for predictive modeling) was developed and applied to five-minute interval wind speed data in Pontianak, West Kalimantan. The dataset, comprising 3,742 observations recorded from December 11 to 24, 2024, was utilized to evaluate the effectiveness of the KF in enhancing model fitting. The model quality was further assessed using Individual Moving Range (IMR) control charts to monitor residual stability and detect localized anomalies. Results demonstrate that the KF-ARIMA approach significantly improves performance, reducing the Root Mean Square Error (RMSE) from 1.123 m/s to 0.145 m/s, representing an 87.1\% improvement in precision compared to the standalone ARIMA model. The I-MR charts confirmed that the KF-ARIMA residuals remained consistently within the $3\sigma$ control limits, effectively identifying transient variations that standard diagnostic tests might overlook. This integrated framework proves that combining state-space filtering with traditional time-series models provides a robust approach for characterizing high-frequency meteorological data in equatorial regions.

Accurate survival prognosis is critical for clinical decision-making, yet analyzing censored on cological data remains a statistical challenge. This study aims to implement the Expectation Maximization (EM) algorithm for the two-parameter exponential distribution to estimate parameters and forecast extreme survival risks in uterine leiomyosarcoma (uLMS). Using clinical data from 122 patients, the EM algorithm achieved rapid convergence after 19 it erations, yielding a scale parameter of 68.7037 months and a 2-month survival threshold. Statistical validity was confirmed by a Kolmogorov-Smirnov test (p = 0.2084) and a 94.5% coverage probability from Monte Carlo simulations. A key contribution of this research is the integration of Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) metrics, which identified a 95% survival threshold of 207.82 months. Sensitivity analysis further demonstrated the es timator’s structural stability across varying censoring proportions. These results signify that the proposed framework provides a robust and reliable tool for quantifying extreme survival probabilities, offering clinicians a sophisticated method for long-term risk management and personalized patient prognosis.