Search Results (183)

DBSCAN is widely used to identify structured regions in unlabeled data, but its performance depends critically on the selection of the neighborhood parameter $\epsilon$. Traditional heuristics for estimating $\epsilon$ often become unreliable in high-dimensional or varying-density settings because they rely heavily on local geometric criteria and may fail under smooth transitions or topological ambiguity. This work presents a three-level perspective on DBSCAN hyperparameter selection. At the algorithmic level, $\epsilon$ controls neighborhood connectivity and structural transitions in clustering. At the modeling level, the ordered k-distance signal is approximated through a surrogate dynamical estimation framework inspired by a mass–spring–damper system. At the causal level, the resulting estimator is interpreted through interventions on its internal threshold-selection mechanism. The proposed method models the variation of $\epsilon$ using ordinary differential equations defined on the ordered k-distance signal, enabling analysis of structural transitions in density organization via a surrogate dynamical representation. System identification is performed using L-BFGS-B optimization on the smoothed k-distance curve, while the system dynamics are solved with the fourth-order Runge–Kutta method. The resulting estimator identifies transition regions that are structurally informative for $\epsilon$ selection in DBSCAN. To analyze the estimator at the intervention level, Pearl’s do-calculus is used to compute the Average Causal Effect (ACE). The method was evaluated on synthetic benchmarks and on the Covtype dataset, including scenarios with multi-density overlap and dimensionality up to ${\mathbb{R}}^{10}$. The resulting ACE values, $+0.9352$, $+0.5148$, and $+0.9246$, indicate that the proposed estimator improves intervention-based $\epsilon$ selection relative to the geometric baseline across the evaluated datasets. Its practical computational cost is dominated by nearest-neighbor search, behaving approximately as $O(NlogN)$ under favorable indexing conditions and degrading toward $O\left(N^2\right)$ in high-dimensional or weak-pruning regimes. Full article

We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on ${[0,1]}^5$, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters. Full article

This paper investigates an optimal cooperative guidance strategy for the active defense of an early-warning aircraft (EWA) escorted by two fighters against an incoming missile. The proposed framework extends classical three-body defense models (Target–Missile–Interceptor) into a more realistic four-body engagement (Target–Missile–Interceptor 1–Interceptor 2), allowing explicit coordination among multiple defenders. By projecting the 3D engagement kinematics onto two orthogonal 2D planes—a validated simplification for typical aerial combat geometries—a tractable dynamic model is obtained. Within this model, an analytical cooperative guidance law is derived using optimal control theory and the calculus of variations, minimizing a multi-objective cost function that combines miss distance, control effort, intercept geometry, and coordination terms. Extensive Monte Carlo simulations across 23 attack directions and multiple initial ranges demonstrate that the proposed method achieves an interception success rate of 99%, with an average miss distance of below 5 m. Robustness tests further confirm stable performance under target maneuver uncertainty, sensor noise, and modeling deviations. The algorithm features closed-form control commands with low computational complexity, enabling real-time onboard implementation. Full article

This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of metric-affine gravity (MAG) and the stochastic variational method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schrödinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems. Full article

Fractional-order models are widely recognized for their ability to capture memory and hereditary effects in biological and physiological systems. In this paper, we develop and analyze a Caputo fractional-order dynamical model for the regulation of blood oxygen saturation (SpO₂) under bounded control inputs. The model incorporates nonlinear saturation mechanisms and auxiliary state variables to represent delayed oxygen transport and adaptation effects. By reformulating the system as an operator equation in a suitable Banach space, sufficient conditions for existence and uniqueness of solutions are established using fixed-point theory. An optimal control problem is then formulated to steer oxygen saturation toward a prescribed safe target level, and the existence of an optimal control is proved via compactness arguments and the direct method of the calculus of variations. Numerical simulations are provided to illustrate the theoretical findings and to demonstrate the impact of the fractional order on transient oxygen saturation dynamics, including comparison with the classical integer-order case. The results show that fractional modeling offers a mathematically rigorous and physiologically interpretable framework for describing delayed oxygenation responses and achieving stable regulation under bounded control constraints. Full article

Gravitational fingering often occurs for water flow in the vadose zone and accurate modeling of this important flow process remains a significant scientific challenge. This paper presents the latest theoretical developments of the optimality-based active region model (ARM), a macroscopic framework developed for describing gravitational fingering flow in the vadose zone. ARM divides the soil into active (fingering) and inactive regions, introducing a relationship between water flux and hydraulic gradient derived from the principle of optimality that the system self-organizes to maximize water flow conductivity. Unlike traditional models, ARM’s hydraulic conductivity depends on both capillary pressure or water saturation and water flux, reflecting the unstable nature of fingering flow. The paper provides an updated mathematical derivation of ARM relationships using calculus of variations and extends ARM to account for small water flux in the non-fingering zone, resulting in a dual-flow field model. These new developments should make ARM more rigorous and realistic for field-scale applications. Full article

In this work, we derive necessary optimality conditions for a class of fractional variational problems involving Caputo-type derivatives. We consider functionals defined on appropriate spaces of absolutely continuous functions and study both periodic and antiperiodic boundary conditions, treated in a unified framework. The analysis covers the cases $0<\alpha <1$ and $1<\alpha <2$, leading to fractional Euler–Lagrange equations supplemented by suitable transversality conditions. We further extend the results to problems with integral constraints and holonomic constraints, as well as to a fractional Herglotz variational principle. Full article

This paper develops an operator-oriented framework for spectral approximation in fractional calculus by introducing a fractional inner product defined through the Riemann-Liouville integral. Instead of modifying polynomial families, the proposed approach continuously deforms the underlying Hilbert space structure, with the fractional order $\alpha$ acting as a deformation parameter. A central theoretical result shows that this fractional inner product is mathematically equivalent to a classical weighted inner product with a deformed weight $w_{\alpha }\left(x\right)={(b-x)}^{\alpha -1}w\left(x\right)$. This equivalence establishes a rigorous connection between fractional calculus and classical orthogonal polynomial theory and clarifies the structural role of the fractional parameter. For a canonical one-dimensional setting, explicit recurrence relations are derived and the limiting behavior as $\alpha \to 1$ is characterized, recovering the classical theory. The resulting orthogonal systems are naturally compatible with fractional operators and are used to construct spectral Galerkin methods for fractional differential equations. Well-posed variational formulations and optimal convergence rates are established. Numerical experiments illustrate the effectiveness of the framework, demonstrating spectral accuracy and improved performance in the approximation of fractional integrals and selected fractional differential equations when compared with standard polynomial bases. The proposed formulation provides a unifying operator-level perspective for spectral methods in fractional calculus. Full article

Dynamic edge-server allocation in mobile edge computing (MEC) networks is a challenging multi-objective optimization problem due to highly dynamic user demands, spatiotemporal traffic variations, and the need to simultaneously minimize service latency and workload imbalance. Existing heuristic and metaheuristic-based approaches for this problem often suffer from premature convergence, limited exploration–exploitation balance, and inadequate adaptability to dynamic network conditions, leading to suboptimal edge-server placement and inefficient resource utilization. Moreover, most existing methods lack memory-aware search mechanisms, which restrict their ability to capture long-term system dynamics. To address these limitations, this paper proposes a Fractional-Order Multi-Objective African Vulture Optimization Algorithm (FO-MO-AVOA) for dynamic edge-server allocation. By integrating fractional-order calculus into the standard multi-objective AVOA framework, the proposed method introduces long-memory effects that enhance convergence stability, search diversity, and adaptability to time-varying workloads. The performance of FO-MO-AVOA is evaluated using realistic MEC network scenarios and benchmarked against several well-established metaheuristic algorithms. Simulation outcomes reveal that FO-MO-AVOA achieves 40–46% lower latency, 38–45% reduction in workload imbalance, and up to 28–35% reduction in maximum workload compared to competing methods. Extensive experiments conducted on real-world telecom network data demonstrate that FO-MO-AVOA consistently outperforms state-of-the-art multi-objective optimization algorithms in terms of convergence behaviour, Pareto-front quality, and overall system performance. Full article

This study presents a novel softsign-function-based fractional-order proportional–integral–derivative (softsign-FOPID) controller optimized using the fungal growth optimizer (FGO) for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces a smooth nonlinear softsign function into the conventional FOPID structure to limit abrupt control actions and improve transient smoothness while preserving the flexibility of fractional dynamics. The FGO, a recently developed bio-inspired metaheuristic, is employed to tune the seven controller parameters by minimizing a composite objective function that simultaneously penalizes overshoot and tracking error. This optimization ensures balanced transient and steady-state performance with enhanced convergence reliability. The performance of the proposed approach was extensively benchmarked against four modern metaheuristic algorithms (greater cane rat algorithm, catch fish optimization algorithm, RIME algorithm and artificial hummingbird algorithm) under identical conditions. Statistical analyses, including boxplot comparisons and the nonparametric Wilcoxon rank-sum test, demonstrated that the FGO consistently achieved the lowest objective function value with superior convergence stability and significantly better (p < 0.05) performance across multiple independent runs. In time-domain evaluations, the FGO-tuned softsign-FOPID exhibited the fastest rise time (0.0089 s), shortest settling time (0.0163 s), lowest overshoot (4.13%), and negligible steady-state error (0.0015%), surpassing the best-reported controllers in the literature, including the sine cosine algorithm-tuned PID, logarithmic spiral opposition-based learning augmented hunger games search algorithm-tuned FOPID, and manta ray foraging optimization-tuned real PIDD². Robustness assessments under fluctuating reference trajectories, actuator saturation, sensor noise, external disturbances, and parametric uncertainties (±10% variation in resistance and inductance) further confirmed the controller’s adaptability and stability under practical non-idealities. The smooth nonlinearity of the softsign function effectively prevented control signal saturation, while the fractional-order dynamics enhanced disturbance rejection and memory-based adaptability. Overall, the proposed FGO-optimized softsign-FOPID controller establishes a new benchmark in nonlinear magnetic levitation control by integrating smooth nonlinear mapping, fractional calculus, and adaptive metaheuristic optimization. Full article

In this paper, we establish and prove two main results: (i) a Kalman-like controllability criterion, and (ii) a rank condition on the controllability matrix, defined via the discrete Mittag–Leffler function, for time-invariant linear fractional-order h-discrete systems. Using some properties of the Mittag–Leffler-type function within the framework of fractional h-discrete calculus, we state and prove the variation of constants formula for an initial value problem. Then we use this formula to prove the equivalence between two notions of controllability: complete controllability and controllability to the origin. Full article

To address the challenges faced by multiple autonomous underwater vehicles (AUVs) in formation control under complex marine environments—such as model uncertainties, external disturbances, dynamic communication topology variations, and limited communication resources—this paper proposes an integrated control framework that combines robust individual control, distributed cooperative formation, and dynamic event-triggered communication. At the individual control level, a robust control method based on a fractional-order sliding mode observer (FOSMO) and a fractional-order terminal sliding mode controller (FOTSMC) is developed. The observer exploits the memory and broadband characteristics of fractional calculus to achieve high-precision estimation of lumped disturbances, while the controller constructs a non-integer-order sliding surface with an adaptive gain law to guarantee finite-time convergence of tracking errors. At the formation coordination level, a distributed trajectory generation method based on dynamic consensus is proposed to achieve reference trajectory planning and formation maintenance in a cooperative manner. At the communication level, a dynamic-threshold event-triggered mechanism is designed, where the triggering condition is adaptively adjusted according to the state errors, thereby significantly reducing communication load and energy consumption. Theoretically, Lyapunov-based analysis rigorously proves the stability and convergence of the closed-loop system. Numerical simulations confirm that the proposed method outperforms several benchmark algorithms in terms of tracking accuracy and disturbance rejection. Moreover, the integrated framework maintains precise formation under communication topology variations, achieving a communication reduction rate exceeding 65% compared to periodic protocols while preserving coordination accuracy. Full article

This work explores the issue of identifying sub-synchronous oscillations (SSOs). Regular detection techniques face issues with response timings to variations in viewpoint and adaptability to variations in conditions of the system but our proposed method overcomes them. We have actually come up with a new framework called Tempered Fractional Deterministic Learning (TF-DL) that successfully combines tempered fractional calculus with deterministic learning theory. This method makes a memory-based learner that works best for oscillatory dynamics. This lets SSO identification happen faster through a recursive structure that can run in real time. Theoretical analysis validates exponential convergence in the context of persistent excitation. Simulations show that detection time is 62.7% shorter than gradient descent, with better convergence and better parameters. Full article

Fractional calculus provides powerful tools for modeling nonlocality, dissipative systems, and, when defined in the time representation, provides an interesting memory effect in mathematical physics. In this paper, we review four standard fractional approaches: the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, and Riesz formulations. We present their definitions, basic properties, Weyl–Marchaud, and physical interpretations. We also give a brief review of related operators that have been used recently in applications but have received less attention in the physical literature: the fractional Laplacian, conformable derivatives, and the Fractional Action-Like Variational Approach (FALVA) for variational principles with fractional action weights. Our emphasis is on how these operators are, and can be, applied in physical problems rather than on exhaustive coverage of the field. This review is intended as an accessible introduction for physicists working in diverse areas interested in fractional calculus and fractional methods. For deeper technical or domain-specific treatments, readers are encouraged to consult the works in the corresponding fields, for which the bibliography suggests a starting point. Full article

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings. Full article