AppliedMath

Complex Network Analysis (CNA) models river systems as networks of nodes (junctions) and edges (channels) to quantify structure and potential transport pathways. Unlike conventional river-network characterization, which often relies on multiple metric-specific analyses of stream hierarchy, branching, drainage density, and topographic characteristics, a graph-based approach provides a unified mathematical framework to assess the entire structural topology and connectivity of the basin simultaneously. In this study, the river networks of Leyte Island were analyzed using 5 m-resolution Interferometric Synthetic Aperture Radar (IfSAR)-derived Digital Terrain Model (DTM) data. River networks were represented as upstream-to-downstream direction graphs, with flow direction assigned based on elevation, and key network parameters, including the number of nodes, average degree centrality, closeness centrality, clustering coefficient, and betweenness centrality, were computed. Simple river networks consisting of two nodes were dominant across Leyte Island, while relatively few large and structurally complex networks were identified. The network metrics generally indicated tree-like or dendritic topologies. Comparison with randomly generated tree networks revealed both similarities and differences in several network metrics across different network sizes. Although both network types exhibit tree-like structures and consequently have zero average clustering coefficients, the differences in other metrics suggest that river networks may exhibit structural characteristics that are not fully represented by randomly generated tree networks. Slope analysis revealed a significant but weak negative correlation between river-network slope and the number of nodes. A key strength of CNA is its data-driven integration of topographic information into the network design, such as slope and elevation, enabling consistent and reproducible characterization of river networks. Future work could further enhance the accuracy of river-network representation through additional spatial validation using higher-resolution geospatial data. Full article

Quantile functions provide a natural representation of probability distributions and offer direct advantages for random variate generation, simulation, and quantile-based statistical analysis. In this paper, we develop a quantile-differential construction in which a probability model is characterized through a differential equation satisfied directly by its quantile function. Within this framework, we introduce a one-parameter continuous model, termed the quadratic quantile distribution (QQD), whose explicit quantile representation leads to a simple and analytically tractable probability model. Closed-form expressions are obtained for the cumulative distribution function, probability density function, survival function, hazard rate, and cumulative hazard function. Several structural properties are also investigated, including shape characteristics, stochastic ordering, entropy measures, fractional moments, and quantile-based descriptive measures. Particular attention is devoted to the upper-tail behavior. The survival function is regularly varying with index $-1$, so that the QQD has a Pareto-type tail with tail index one. Consequently, the ordinary mean is infinite, whereas fractional moments exist only for orders less than one. This feature makes quantile-, survival-, and tail-based summaries more appropriate than conventional moment-based descriptions. Maximum likelihood estimation is developed for the model parameter, and its finite-sample behavior is examined through an extensive Monte Carlo study. The simulation results show a progressive reduction in bias, root mean squared error, and mean absolute error as the sample size increases, providing numerical evidence consistent with the expected large-sample behavior of the estimator without treating simulation as a proof of asymptotic consistency. The empirical performance of the QQD is examined using three right-skewed datasets from reliability and biomedical applications. Comparisons with established one- and two-parameter distributions show that the QQD can provide a competitive likelihood-based fit while retaining the simplicity of a single unknown parameter. Graphical comparisons further complement the numerical criteria, particularly in assessing the behavior of the fitted models in the upper tail. Overall, the results suggest that the QQD offers a parsimonious and analytically convenient alternative for selected strongly right-skewed positive data, while its suitability should be assessed on a case-by-case basis in view of its heavy-tailed structure. Full article

The mathematical analysis of the hyperbola is extended to distinguish and compare several angles which are important to physical applications, e.g., comets and Rutherford scattering. We quantify the impact angle, which plays a key role in angular momentum conservation, and distinguish it from the angle of deflection emphasized in historical analyses of gravitational bending of light. We elucidate requirements for collinear arrangement of points on opposite limbs with the focus, which configuration pertains to analysis of galaxy images for possible gravitational lensing effects. These geometrical relationships are obtained by transforming the curved trajectory of a mass propelled by Newtonian gravitational forces in polar (R, θ) or conventional Cartesian (x, y) Euclidean space into linear trajectories in Y, R or X, R space, where R, X, and Y are radial and Cartesian distances to the focus, now located at X = 0 and Y = 0. These coordinate transformations simplify geometric analysis of orbits and trajectories, as they allow any conic section to be recast into the compact linear form, R = eY + L, where e is the standard eccentricity and L is the length of the semi-latus rectum. Our formulae specify the limitations of approximations used in physical analyses of light bending, such as considering only asymptotic behavior and small angular deviations. Here we quantitatively describe the entire hyperbolic path, which should be useful in many applications. Full article

This paper develops a multidimensional differential-transform-based framework for computing matrix characteristics of complex-valued multiparameter matrix functions. The proposed approach extends differential-transform techniques from one-parameter matrix functions to functions depending on several independent variables and constructs multidimensional D-analogues of the Leverrier and Faddeev methods. In the spectral domain, products of parameter-dependent scalar and matrix functions are replaced by multidimensional convolutions, which makes it possible to compute the spectra of characteristic-polynomial coefficients, determinants, and inverse-matrix entries by recurrence relations. The convergence of the inverse multidimensional transform is discussed in terms of analyticity in a polydisc, and truncation-error estimates and residual-based a posteriori indicators are introduced for controlling the accuracy of reconstructed inverse matrices in locally nonsingular parameter regions. The method is implemented in a Python 3.14-based computational framework that stores and manipulates multidimensional differential spectra. The approach is verified on multiparameter complex matrix examples, including a three-degree-of-freedom damped vibration system and a dense $7 \times 7$, seven-parameter complex matrix. Numerical residuals confirm the consistency of the reconstructed inverse matrices with MATLAB R2025b-based verification. The results show that the proposed recurrence-based method is especially useful for sparse multidimensional spectra, bounded-order local reconstructions, and repeated evaluations of matrix characteristics near a fixed expansion point. Full article

We investigate the convergence of the Levenberg–Marquardt (LM) iterative regularization method for linear ill-posed inverse problems in Hilbert spaces under general source conditions characterized by admissible index functions. We introduce admissibility conditions tailored to the spectral filter structure of the LM iteration, extending the classical convergence theory beyond Hölder-type source conditions to a broader class of index functions, including logarithmic source conditions relevant to severely ill-posed inverse problems. Using the filter representation of the LM iteration, we derive convergence rate estimates under both a priori and an a posteriori parameter choice rules. The analysis provides a unified framework for convergence analysis that covers a broad class of admissible index functions. Numerical experiments involving a time-fractional backward heat problem and a Fredholm integral equation of the first kind are consistent with the theoretical convergence results and demonstrate close agreement between the predicted error estimates and the observed reconstruction errors under both Hölder-type and logarithmic source conditions. Full article

Malaria and typhoid fever co-infection produces severe illness affecting public health, especially in countries where both diseases coexist. A mathematical model that considers the critical factors influencing the transmission dynamics and control interventions of malaria and typhoid fever co-infection is developed. The essential epidemiological features of the model, such as the basic reproduction number and disease-free equilibrium, are determined and analysed. A dynamical systems analysis of the model reveals the conditions under which the disease can be eradicated or persists. Numerical simulations are conducted using real data from Nigeria as a case study. By fitting the model to the real data, essential parameter values are estimated and model prediction that reveals the possible long-term dynamics of the model is determined. The impact of the various control interventions are investigated. These findings are anticipated to aid in improving the management of malaria–typhoid co-infection in endemic regions for expeditious disease eradication. Full article

Lumpy skin disease (LSD) is a rapidly spreading viral disease of cattle that threatens livestock productivity and agricultural trade across Africa, the Middle East, and Asia. Despite effective vaccines being available, evidence-based guidance on allocating limited veterinary budgets across competing interventions remains lacking, and no LSD study has yet combined dynamic transmission modelling with a hard budget constraint and formal cost-effectiveness analysis. We develop a deterministic $S_h{V}_h{E}_h{I}_h{R}_h$–$S_v{I}_v$ (SVEIRSI) compartmental model for LSD, embedded within a budget-constrained optimal control framework with three time-dependent controls—vaccination ($u_1$), vector control ($u_2$), and movement restriction ($u_3$)—evaluated across all eight strategy combinations. The basic reproduction number is ${\mathcal{R}}_0=2.7899$. Pontryagin’s Maximum Principle characterises the optimality conditions. The resulting two-point boundary value problem is solved via forward–backward sweep with budget bisection over a 180-day horizon at three budget levels (${\mathcal{W}}_L=\mathrm{NAD}5000$, ${\mathcal{W}}_M=\mathrm{NAD}\mathrm{15,000}$, ${\mathcal{W}}_H=\mathrm{NAD}\mathrm{40,000}$). Vaccination dominates the optimal allocation at every budget, reducing peak infectious prevalence by 54–57% and averting 15,645–17,970 animal-days of cumulative burden relative to the uncontrolled baseline (${\mathcal{I}}_0=\mathrm{97,313}$ animal-days). Shadow-price analysis reveals that the budget constraint is non-binding across all three policy budget levels (actual optimal spend ≈ NAD 572), indicating that logistical (rather than financial) barriers are the true bottleneck, with the shadow price dropping to zero at $\mathcal{W}\approx \mathrm{NAD}1862$. Incremental cost-effectiveness analysis places vaccination alone ($S_1$), vaccination plus vector control ($S_{12}$), and the combined strategy ($S_{123}$) on the efficiency frontier, with ICERs of NAD 0.012, 0.015, and 0.166 per animal-day averted, respectively. One-way sensitivity analysis identifies vaccine efficacy $\epsilon$ as the dominant driver of cases averted (swing $\approx \mathrm{85,601}$ animal-days), while global sensitivity analysis via Latin Hypercube Sampling identifies the midge biting rate a as the strongest driver of cumulative infectious burden ($\mathrm{PRCC}=0.926$, $p<0.001$). These results provide actionable, quantitative guidance for LSD outbreak response in Namibia and comparable resource-limited settings. Full article

Low adult density can suppress recruitment through mate limitation or cooperative reproductive failure, even when a host improves adult survival. Motivated by this ecological tension, we introduce a fecundity-related component Allee effect into a stage-structured commensalism model with a Holling type II host benefit. The nonlinear recruitment term is $f\left(x_2\right)=\alpha x_2^2/(x_2+A)$, and the independently growing host converges to its carrying capacity, yielding an asymptotically autonomous commensal subsystem. We obtain a complete analytical classification. If the long-term host benefit D exceeds mature mortality ${\delta }_2$, a unique coexistence equilibrium attracts every nontrivial commensal population. If the benefit is insufficient, fecundity limitation can instead create two positive equilibria: a low-density saddle whose stable set is the basin boundary and a stable high-density coexistence state. Thus, extinction and coexistence are both possible, depending on initial stage composition and host abundance. Stronger fecundity limitation enlarges the extinction region, whereas greater recruitment, maturation or host benefit promotes persistence. At the critical boundary, the positive equilibria merge in a non-degenerate saddle-node bifurcation. Rigorous global arguments and numerical comparisons with the linear-recruitment model show how a component Allee effect can induce strong-Allee-type demographic behavior, including bistability and threshold-mediated extinction. Full article

This paper addresses loop-wise PI/PID controller structure selection in a coupled two-tank hydraulic system. Existing studies generally prescribe controller architectures or compare predefined configurations after synthesis, leaving this problem insufficiently explored when each choice depends energetically on the neighboring loop. Although exact feedback linearization decouples the auxiliary tracking dynamics, the reconstructed physical pump inputs retain nonlinear hydraulic coupling. Consequently, reducing the control energy of one loop may increase the effort required by the other, creating conflicting local objectives that motivate a non-cooperative game-theoretic formulation. The plant is feedback-linearized, PI and PID controllers are parameterized by pole placement, and their assignment is posed as a finite static game in which each player minimizes pump energy cost. For the nominal symmetric configuration, (PID, PID) is the unique Nash equilibrium, although (PI, PI) yields a lower aggregate control energy cost (291.4134 vs. 291.8184), showing that unilateral optimality does not coincide with collective energy minimization. PID reduces settling time from 5.253 s to 3.422 s (34.9%) but increases maximum overshoot from 13.55% to 20.82% (53.7%). Sensitivity analyses show that tuning, leakage, coupling, and initial conditions can reverse best responses, while a $\pm 10\%$ parametric mismatch analysis preserves convergent tracking with small steady-state errors, although the equilibrium may change. Full article

We develop a predictive–prescriptive decision-support framework for football squad composition under budget and positional constraints, in which mixed-integer optimization is the central methodological contribution. The framework integrates four components: (i) construction of a Sports Performance Index (SPI) via principal component analysis (PCA) of standardized per-90 performance features; (ii) leakage-free, player-wise, five-fold cross-fitted prediction of seasonal goals plus assists using ordinary least squares (OLS) as the primary input model, regularized linear models as additional baselines, and gradient boosting (XGBoost) as a nonlinear benchmark and uncertainty-estimation component, with each player-season receiving an out-of-fold prediction; (iii) a mixed-integer linear program (MILP) that selects a player-season portfolio under budget, position composition, squad-size, and duplicate-player constraints, with a tunable weight balancing predicted attacking output against the performance index; and (iv) a robust counterpart that incorporates prediction uncertainty into the optimization. The framework is implemented on English Premier League data covering the 2023–2024 and 2024–2025 seasons (N = 218 forward and midfielder player-seasons with at least 1800 min of playing time). Cross-fitted OLS attains a five-fold mean R² = 0.76 ± 0.03 with stable performance across folds. The exact MILP outperforms a benefit-to-cost greedy heuristic by up to 9.4% and the best of 1000 random feasible rosters by 41.0–53.0% in objective value, while guaranteeing feasibility of all composition and budget constraints. Sensitivity analyses across budget tightness, position bounds, squad size, and cost proxy demonstrate qualitative robustness. The robust counterpart with risk-aversion parameter κ ∈ {0, 0.5, 1.0} produces a Jaccard roster overlap of 0.75 relative to the nominal solution and reduces average selected-player prediction uncertainty from 2.65 to 2.49. The contribution is a mathematically rigorous, integer-programming-centered decision-support framework that integrates machine learning with optimization for sports business analytics and is directly applicable to club-level squad planning. Full article

This work introduces efficient signal denoising methods based on wavelet thresholding (WT) and Hankel matrix-based Singular Value Decomposition (HSVD) with a randomized algorithm. The sequential hybrid frameworks of these techniques are investigated—both WT followed by HSVD (WT-HSVD) and HSVD followed by WT (HSVD-WT)—across varying noise levels. Numerical tests are performed through different benchmark signals, including dual harmonic, damped sine, and dual frequency signals, as well as a natural phonocardiogram recording. Different types of corrupted noises, including white Gaussian, colored (brown), and impulsive noises, are considered in the numerical tests. For moderate and high noise levels, hybrid frameworks can significantly improve the final accuracy of the denoised signals, as measured by reconstruction error metrics and Signal-to-Noise Ratio. These hybrid frameworks are also shown to be advantageous in compensating for suboptimal parameter selections that may occur when using either method individually. Furthermore, to mitigate the computational burden inherent to exact SVD, this work employs a randomized Singular Value Decomposition (rSVD) algorithm. Depending on the signal size, the proposed hybrid frameworks with rSVD achieve a 30% to 80% reduction in CPU execution time while maintaining denoised signal reconstruction accuracy across test cases. Full article

Friction stir welding (FSW) is a solid-state manufacturing process widely used in joining aluminum and other metal workpieces. The FSW process can be modeled by a coupled system of non-Newtonian Navier–Stokes and heat-transfer equations. However, solving this non-linear system with high accuracy requires significant computational power. This work refines the system by introducing corrected coefficients and new treatments for boundary conditions near the tool. Then, model order reduction, including the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), is applied to efficiently solve the FSW system in a low-dimensional space. To enhance accuracy and effectiveness, two novel treatments have been adopted for the POD and DEIM models: the introduction of preconditioner matrices to avoid large condition numbers and the use of indicator matrices to generate the non-linear data. For different cases regarding operating parameters, the results show that the DEIM model dramatically speeds up the computation (e.g., over 250 times faster compared with the full model) while maintaining accuracy. This makes simulations of the FSW process more accessible and easily combined with machine learning techniques in future study. Full article

Mastitis remains one of the most important infectious diseases affecting dairy cows, with substantial consequences for animal health, milk production, and farm profitability. In this study, we extend a nonlinear SIRS–P compartmental model for mastitis transmission in dairy cows by incorporating sensitivity analysis, optimal control, and cost-effectiveness analysis. The model consists of susceptible, infected, and recovered cow populations together with an environmental pathogen compartment, and accounts for direct cow-to-cow transmission, indirect transmission through environmental contamination, recovery, recurrence of infection, recruitment, and culling. A sensitivity analysis of the basic reproduction number $R_0$ is performed to identify the parameters that most strongly influence mastitis transmission. An optimal control problem is then formulated using time-dependent controls that reduce direct transmission, reduce environmental contamination, and enhance recovery. Pontryagin’s Maximum Principle is applied to derive the optimality system, which is solved numerically using the forward–backward sweep method. Seven intervention strategies are compared through numerical simulations and cost-effectiveness analysis using ACER and ICER. The results indicate that the fully integrated strategy $S_7$, which combines prevention and screening, environmental sanitation, and treatment, provides the largest overall disease reduction. However, Strategy $S_1$, based on direct transmission reduction, offers the most cost-effective allocation of resources for reducing the mastitis infection burden. Full article

This paper applies the updated fuzzy model proposed by Belhadj and Bounani in 2023 to microdata from the National Household Survey (ENAHO) for the province of Trujillo, La Libertad, Peru, to measure monetary poverty. The study demonstrates that this framework offers a more detailed, continuous assessment of poverty levels than traditional binary classifications. By assigning degrees of deprivation to various poverty indicators, the fuzzy approach captures the full intensity of deprivation, uncovering intermediate socioeconomic states that conventional methods systematically overlook. While the official binary classification by INEI categorizes 94.62% of Trujillo’s population as “non-poor,” the fuzzy methodology reveals that only 22.18% experience complete non-deprivation. Crucially, it exposes that 49.94% of the population resides in a state of “mild poverty”—a highly vulnerable segment rendered invisible by standard metrics. District-level disaggregation highlights sharp territorial disparities, ranging from severe structural vulnerability in Florencia de Mora (Fuzzy Poverty Index = 0.428) to consolidated stability in Víctor Larco Herrera (0.288). Ultimately, the fuzzy approach provides a more nuanced and realistic representation of poverty dynamics. This offers crucial insights for policymakers designing targeted poverty-reduction interventions. Full article

Deep mining operations in metal mines face escalating challenges from high in situ stress environments, where conventional single-method ground control approaches often prove insufficient. This study proposes an innovative combined protection technology integrating four synergistic components: (i) prestressed rock bolt and plate support for immediate roadway reinforcement, (ii) hydraulic fracturing-based shielding to create fracture zones isolating the orebody from surrounding high-stress regimes, (iii) destress blasting at the orebody crown to form a stress-transfer barrier, and (iv) subsequent cemented paste backfill to provide long-term regional stability. Theoretical formulations are derived for each component: a bolt–rock composite bearing model quantifies the confinement provided by prestressed support; a fracture mechanics-based model characterizes the stress-shielding efficiency of hydraulic fracture networks; controlled blasting theory predicts the stress redistribution achieved through crown destressing; and a backfill arching model evaluates the long-term load-bearing capacity of cemented paste fill. A three-dimensional FLAC3D numerical model was established to simulate six working conditions: no protection, support-only, shield-only, pressure relief-only, backfilling-only, and the integrated four-component system combining support, shield, pressure relief and backfilling. The numerical results reveal that the integrated composite protection system reduces roadway subsidence by 86%. Furthermore, the synergistic effect among all components exceeds the sum of individual contributions, which verifies the necessity of adopting the comprehensive protection strategy under deep high in situ stress conditions. The research findings provide theoretical basis and engineering design references for safe and efficient mining in deep metal mines. Full article

A delayed malware dissemination model for wireless sensor networks is extended by replacing the single fixed anti-virus cycle time with distributed recovery memories. Two normalized gamma kernels are considered: a one-stage exponential memory (weak kernel) and a two-stage gamma memory (strong kernel), both representing heterogeneous anti-virus removal times across carrier and infectious nodes. The linear chain trick transforms the corresponding integro-differential systems into six- and eight-dimensional ordinary differential equation models. For these distributed-delay formulations, we derive the common disease-free equilibrium and its local stability conditions, prove that the endemic equilibrium and the basic reproduction number are preserved, and show that the inherited delayed-removal mechanism does not generate a positively invariant nonnegative orthant; in its place, we establish the strongest valid conditional boundedness and continuation result. Local endemic stability is reduced to a Routh–Hurwitz problem, and Hopf thresholds are obtained from simple purely imaginary roots together with transversality. For the reference parameter set, full spectral verification and first-Lyapunov-coefficient evaluation show that the first weak- and strong-kernel Hopf points are subcritical. Numerical simulations illustrate convergence below threshold and loss of equilibrium stability above it, while comparison with the discrete-delay benchmark shows that a broader recovery-time distribution postpones instability. Full article

Large language models (LLMs) have recently improved their problem-solving abilities and can solve complex mathematical problems with an increasing accuracy, necessitating the development of more challenging benchmarks. Over the years, the performance of LLMs on several benchmark datasets has also improved, motivating the development of evaluation frameworks that emphasize deeper algorithmic reasoning. This research presents Bench of Euler, a benchmark comprising 954 challenging mathematical and computational puzzles from the “Project Euler” collection, designed to evaluate complex multi-step mathematical reasoning. Solving these puzzles requires in-depth algorithmic deduction, as solutions based on brute force enumeration are mathematically infeasible. The benchmark is further classified into bands with respect to the problem difficulty to enable structured evaluation. Several existing open-source and proprietary, as well as base and reasoning, models have been evaluated on Bench of Euler, among which the highest-performing gpt-4o model by OpenAI achieved the highest accuracy of 43.21%. To evaluate the algorithmic efficiency, the models were prompted to generate Python code, which was subsequently executed in an isolated sandbox with strict timeout constraints. A failure analysis of the top-performing models revealed their inability to effectively optimize solutions, resulting in time-limit exceeded (TLE) errors during execution and outputs that often remained incomplete or approximate. Additionally, second-pass evaluations yielded an average performance improvement of approximately 2.09%, indicating limited gains from a second stochastic sample under the adopted decoding protocol. Full article

In this work, we prove the strong convergence of an inertial iterative scheme to approximate solutions of the equilibrium fixed point problem associated with nonexpansive mappings in Hilbert spaces. Numerical simulations are carried out to examine the performance of the proposed approach. The results indicate that the proposed inertial approach achieves faster convergence when compared with existing comparable iterative schemes. We also investigate how different choices of initial values influence the convergence behavior of our algorithms and we compare these effects with those observed in classical iterative schemes through graphical illustrations. In applications, we used our approach to solve the signal processing problem. We also apply it to a mathematical model describing the spread of an infectious disease, which illustrates its relevance to real-world dynamical systems. Finally, we show that the proposed method can be applied in solving constrained optimization, variational inequality and split feasibility problems which highlight its flexibility and wide applicability. Full article

We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form ${\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{(\pm 1)}^k}{w_{rk+l}{w}_{mk+s}{w}_{m(k+1)+s}}}$, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter m, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms. Full article