Search Results (100)

Symmetry is crucial in lattice cryptography, where secure signatures rely on structural invariants over polynomial rings. This paper conducts a rigorous security and correctness analysis on Falcon-M, a lightweight signature scheme. We first demonstrate a fundamental correctness failure through an explicit experimental instantiation of the scheme: because the signing algorithm is algebraically decoupled from the secret key, no honestly generated signature was accepted in our tested experiments. Furthermore, we reveal that removing the NTRU trapdoor breaks the essential computational asymmetry, causing the verification equation to degenerate into a publicly solvable linear system. Consequently, for invertible public keys, an adversary can execute a direct universal forgery attack purely from public data via pointwise algebraic inversion in the frequency domain with $\mathcal{O}\left(n\log n\right)$ time complexity. For non-invertible keys, we further identify a practical existential forgery utilizing a localized salt-search. Ultimately, these practical cryptanalytic results mathematically invalidate the claimed security under the analyzed instantiation. Full article

Rough set methodology has gained broad acceptance as a potent mathematical apparatus for feature selection within data mining and machine learning. Yet, classical rough sets hinge on equivalence relations to partition the universe, thereby demanding strict reflexivity, symmetry, and transitivity conditions that are arduous to satisfy in realistic settings. Although fuzzy rough sets have been explored to mitigate this rigidity, the entropy-based uncertainty measures employed in fuzzy approximation spaces remain acutely sensitive to data quality and noise corruption, potentially inducing severe bias in feature evaluation. Moreover, the literature currently lacks noise-tolerant uncertainty measures capable of accommodating a controlled fraction of classification errors while safeguarding the discriminative strength of feature subsets. Inspired by these gaps, this study develops a feature selection framework grounded in variable precision fuzzy entropy within the fuzzy rough set context. To this end, fuzzy decision is adopted to portray the membership degree of samples relative to decision classes, thereby enabling more precise detection and elimination of redundant attributes during approximation. An uncertainty quantifier termed fuzzy relational entropy is then introduced to appraise the distinguishing power of fuzzy similarity relations generated by attribute subsets. Leveraging fuzzy decision, a portfolio of uncertainty measure variants, specifically the variable precision joint discriminant index, the variable precision conditional discriminant index, and the variable precision mutual discriminant index, is developed to counteract noisy data effects. These variable precision discriminant indexes sanction a regulated error proportion and afford a measure of noise resistance. Finally, knowledge reduction for fuzzy decision systems is attacked from the angle of discriminative capability preservation, and a heuristic feature selection algorithm is crafted around the variable precision conditional discriminant index. Evaluation on twelve public UCI datasets reveals that the proposed algorithm effectively prunes redundant features and delivers competitive results against three representative alternatives: classical rough set, neighbourhood-based discriminant index, and fuzzy rough set feature selection. Additionally, it sustains stable classification performance across an extensive sweep of the variable precision parameter. Full article

In industrial trajectory planning and real-time motion control, quintic B-splines are widely used for corner smoothing owing to their local support and high-order continuity. However, existing evaluation methods mainly rely on basis-function recursion or the de Boor algorithm, with limited attention paid to the analytical properties of fixed-topology continuity-constrained structures. This study reveals that, under geometric symmetry and C³ continuity constraints at the junction points, higher-order derivative control-point structures undergo progressive geometric degeneration, whereby second- and third-order derivatives reduce to one-dimensional forms governed by a single direction. Based on this degeneration property, a closed-form reconstruction method for fixed-topology quintic B-spline corner smoothing is developed, yielding unified closed-form expressions for curve position and first- to third-order derivatives. Mathematical analysis proves equivalence between the proposed reconstruction and the original quintic B-spline representation. Numerical validation and efficiency evaluation demonstrate machine-precision consistency with conventional B-spline evaluation while achieving an approximately 3–7-fold speedup in curve and derivative evaluation. System-level trajectory-planning simulations further confirm reduced geometric computation load. The proposed method provides an efficient analytical evaluation framework for real-time trajectory planning and demonstrates how continuity constraints can be exploited to derive efficient analytical spline representations. Full article

This paper addresses a human resource allocation problem with spatial-temporal constraints (HRAP-SC) in the parallel assembly of complex products, such as satellites and aircraft. It involves coordinating a limited pool of multi-skilled workers across geographically distributed workstations, subject to rigorous constraints including team collaboration requirements, operation priorities, technological tail times (e.g., curing), and strict 8 h workdays. Existing exact approaches typically fail to converge due to the combinatorial explosion arising from the strong coupling of shared resources across workstations, while meta-heuristic methods often suffer from performance instability caused by hyper-parameter sensitivity. To overcome these limitations, we propose a pattern-based decomposition algorithm (PDA), a novel parameter-free exact solution framework. By exploiting the inherent symmetry of identical jobs and parallel workstations, PDA defines a set of canonical patterns to drastically reduce the search space. It employs an efficient traversal mechanism reinforced by rigorous mathematical bounds and pruning rules to eliminate unpromising solutions. Computational experiments demonstrate that PDA significantly outperforms state-of-the-art Mixed-Integer Programming (MIP) and Constraint Programming (CP) solvers. Unlike standard solvers, which frequently time out (3600 s), PDA strictly evaluates only a single pattern when proving optimality, and robustly scales to large industrial instances (e.g., six jobs comprising 78 operations) to provide high-quality schedules. By successfully solving complex scheduling problems that remain intractable for monolithic solvers, PDA provides a robust and automated decision-support tool for production management in complex manufacturing systems. Full article

The complexity of a graph, defined as its number of spanning trees, serves as a key measure of network reliability. Stacked graphs constitute a significant and versatile class of graphs, formed by superimposing multiple copies of a base graph upon a shared central vertex set. Their inherent layered symmetry and structural regularity make them compelling models for a wide range of real-world networks, including multi-tier communication systems, hierarchical data networks, and resilient distributed architectures. Moreover, their systematic construction from well-known graph families renders the study of their complexity both mathematically rich and algorithmically meaningful. In this paper, we derive closed-form formulas for the complexity of several stacked graph families based on path- and cycle-based structures with a central vertex, including stacked fan and wheel graphs, stacked double fan and double wheel graphs, and stacked path flower, cycle flower, and gear graphs. The derivations are based on techniques from linear algebra, matrix theory, and Chebyshev polynomials. Full article

This paper presents a visual and quantitative study of Mandelbrot sets generated by the IA-iterative algorithm for the transcendental complex map $T_c\left(z\right)=z^p+sinh\left(c^q\right)$, where $p\ge 2$, $q\ge 1$, and $c\in \mathbb{C}\setminus \left\{0\right\}$. A rigorous escape criterion is derived for the proposed map under the IA iteration scheme, providing the mathematical foundation for the generation and analysis of the associated fractal sets. Using this criterion, a broad family of Mandelbrot sets is constructed to examine the influence of the polynomial degree p and the transcendental parameter q on geometric complexity, symmetry, and branching behavior. To complement the visual analysis, a quantitative computational investigation is performed using the fractal dimension, non-escaping area index, average escape time, and average number of iterations. The numerical results reveal nonlinear relationships between these measures and the iteration parameters $(\alpha ,\beta ,\gamma ,\lambda )$, demonstrating the adaptability of the IA scheme for controlled fractal generation. A comparative analysis with CR iteration is also included to evaluate structural variation, computational behavior, and the broader relevance of the method for computer-based fractal modeling, visualization, and algorithmic design. Full article

Wireless sensor network (WSN) coverage optimization is a challenging high-dimensional and nonlinear problem that directly affects network performance, including sensing quality, energy efficiency, and system reliability. Although metaheuristic algorithms have been widely applied to this problem, many existing methods still suffer from premature convergence, insufficient population diversity, and an imbalance between exploration and exploitation. To address these issues, this paper proposes a multi-strategy enhanced enterprise development optimization algorithm (MEEDOA) inspired by business management mechanisms. The proposed method integrates a hybrid population initialization strategy, an adaptive activity switching mechanism based on performance feedback, a multi-elite collaborative learning strategy, and a Lévy flight-based stagnation escape mechanism. These strategies are tightly coupled within a unified adaptive framework to improve global search capability, convergence speed, and robustness. Furthermore, a mathematical model for WSN deployment is constructed based on a binary sensing model and discrete coverage evaluation. From the perspective of symmetry, the sensing regions of sensor nodes exhibit significant geometric symmetry in both two-dimensional and three-dimensional deployment spaces. In the two-dimensional case, the sensing and communication regions are modeled as concentric circular structures, while in the three-dimensional case, the sensing regions are represented by isotropic spheres with symmetric spatial distributions. Such symmetry properties provide an effective basis for describing coverage behavior, reducing redundant overlap, and improving the uniformity of node deployment. Meanwhile, the proposed MEEDOA preserves population diversity and enhances search balance, enabling the algorithm to better capture symmetric coverage patterns and more effectively explore complex spatial deployment configurations. Extensive experiments on CEC2014, CEC2017, CEC2020, and CEC2022 benchmark functions demonstrate that MEEDOA achieves superior convergence accuracy, faster convergence speed, and stronger robustness compared with several state-of-the-art algorithms. Additional simulation results in WSN deployment applications verify its effectiveness in improving coverage performance under both symmetric and irregular spatial deployment scenarios. The results indicate that the proposed MEEDOA provides a reliable and efficient solution for complex global optimization problems and practical engineering applications. Full article

Air–ground collaborative systems leverage the complementary strengths of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) and hold significant potential for logistics in complex, unstructured environments. However, trajectory planning in infrastructure-free mountainous regions remains challenging owing to the need for continuous tight coupling, obstacle avoidance, and reliable communication-link maintenance. To address these challenges, this study proposes a cooperative trajectory planning framework that enforces strict inter-vehicle distance constraints to maintain communication connectivity. By formulating the coordination problem in terms of relative configurations between air and ground vehicles, the proposed framework exhibits translational invariance, reflecting an underlying symmetry with respect to global position shifts. This symmetry-aware formulation reduces reliance on absolute coordinates and promotes consistent cooperative behavior under environmental variability. The trajectory planning problem is mathematically formulated as a constrained multi-objective nonlinear programming (MONLP) model that balances energy consumption and trajectory smoothness. An adaptive inertia weight particle swarm optimization (AIWPSO) algorithm is developed to efficiently solve the resulting optimization problem. Simulation results demonstrate that the proposed approach generates smooth, collision-free trajectories while maintaining stable air–ground coordination, demonstrating improved feasibility and robustness over conventional planning methods in unstructured mountainous environments. Full article

Structural symmetry appears as a natural feature in both optimal solution landscapes and hospital scheduling behaviors, representing an inherent balance that can be deliberately leveraged to improve how quickly algorithms converge and how reliably systems perform in intricate healthcare optimization contexts. Managing hospital resources is a multifaceted challenge that requires simultaneously addressing several competing goals, such as reducing costs, improving patient experiences, making the most of available resources, distributing staff workload fairly, and strengthening readiness for emergencies. Traditional optimization approaches frequently struggle to cope with the complexity and ever-changing nature of modern healthcare environments. To address this gap, this study introduces a novel Multi-Objective Dream Optimization Algorithm (MO-DOA) tailored for smart healthcare resource management, which adapts a biologically inspired optimization framework to meet the specific demands of healthcare settings. The MO-DOA is built around three core mechanisms: a foundational memory component that retains high-quality solutions, a forgetting-supplementation component that maintains a productive balance between exploration and exploitation, and a dream-sharing component that promotes diversity among candidate solutions. Rigorous testing across realistic hospital environments confirms MO-DOA’s outstanding effectiveness, with results showing a 21.86% gain in resource utilization, a 30.95% decrease in patient waiting times, a 19.06% boost in patient satisfaction, and a 29.56% improvement in how evenly staff workloads are distributed. The algorithm’s emergency response capabilities are especially noteworthy, achieving bed assignments within 4.23 min and an equipment deployment success rate of 94.56%. Computationally, the algorithm proves highly efficient, with an average response time of 18.87 s and strong scalability across different operational scales. Collectively, these findings position MO-DOA as a powerful and practical tool for optimizing hospital operations in real time. Full article

This article presents a comprehensive and analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric. Focusing on highly symmetric configurations on the unit sphere ${\mathbb{S}}^2$, we investigate three explicit models of discrete uniform distributions and derive closed-form expressions for their optimal quantizers and corresponding mean square quantization errors. (I) For N equally spaced points on the equator, we obtain exact error formulas for both divisible and non-divisible cases $n\nmid N$, demonstrating that optimal Voronoi cells form contiguous arcs with midpoint representatives. (II) For two antipodally symmetric small circles at latitudes $\pm {\varphi }_0$, each with M longitudes, we prove a no-cross-circle Voronoi phenomenon, establish symmetry-preserving optimality, and derive finite-sum error formulas together with sharp curvature-dependent bounds and asymptotics. (III) For a single small circle at latitude ${\varphi }_0$, we obtain analogous exact error formulas and show that curvature reduces distortion by a factor of ${cos}^2{\varphi }_0$, while preserving the $n^{-2}$ decay rate. Across all models, we rigorously establish the “block midpoint principle”: optimal Voronoi cells on a circle are contiguous azimuthal blocks, and their optimal representatives are the corresponding azimuthal midpoints. Numerical tables and illustrative figures highlight curvature effects and compare divisible and non-divisible cases. An algorithmic appendix provides pseudocode and a small, commented Python implementation to facilitate reproducibility. Written with didactic clarity while maintaining full mathematical rigor, this work bridges geometric intuition and analytic precision, providing explicit benchmark models that illuminate curvature effects and support further developments in quantization on curved manifolds. Full article

In multi-module boost power factor correction (PFC) systems, current ripple is commonly mitigated by applying fixed 180° interleaving between modules; however, this approach relies on matched inductors and ideal symmetry. In practical implementations, inductor mismatch and duty-cycle variations prevent full cancellation, leading to residual ripple that increases losses and electromagnetic interference. To address this issue, several research works have proposed centralized coordination or high-speed communication among units. However, an explicit converter model is necessary, which makes the system more complicated and expensive. To resolve this problem, this paper presents an extremum seeking optimization method for reducing high-frequency ripple in multi-module PFC systems without requiring explicit converter models. The ripple minimization problem is formulated as a nonlinear, time-varying optimization task, where the relative switching phases of the modules are adaptively tuned. The proposed extremum seeking algorithm perturbs the phase shift, evaluates a ripple-based cost function, and updates the phases iteratively. A harmonic analysis is developed to characterize the dependence of ripple on duty ratio, inductor values, and phase displacement. Simulation results show that the method effectively reduces the RMS ripple current across balanced and mismatched operating conditions. In a three-unit system, applying the proposed technique lowered the current THD to 1.29% compared to 1.44% achieved with a fixed phase-shift approach. These findings demonstrate that extremum seeking optimization provides a mathematically rigorous and practically implementable solution for decentralized ripple minimization in multi-module boost PFC systems. Full article

This study presents the mathematical formulation of a symmetry-compact three-step algorithm (TSA) for the numerical computation of the spatio-temporal generalized FitzHugh–Nagumo equation (FHNE), a class of one-dimensional time-dependent initial-boundary value partial differential equations. The proposed symmetry-compact TSA is constructed using the Lagrange polynomial as the basis function, yielding a structurally balanced and computationally compact formulation with an inherent symmetry that facilitates automatic step-size adaptation over the integration interval. The symmetry-compact nature of the formulation enhances numerical stability while maintaining a reduced computational footprint, thereby improving both accuracy and efficiency when compared with existing numerical schemes. Prior to the application of the TSA, the FHNE is discretized in space, resulting in a system of ordinary differential equations suitable for time integration. Rigorous analyses of the stability and convergence properties of the symmetry-compact TSA are carried out to establish the reliability and robustness of the method. The performance of the proposed algorithm is quantitatively assessed using absolute error, maximum error, root mean square error, and central processing unit time for selected spatio-temporal test cases of the FHNE. The numerical results and corresponding solution profiles clearly demonstrate that the symmetry-compact TSA delivers superior accuracy, enhanced computational efficiency, and improved stability characteristics relative to existing methods, particularly in the presence of stiffness and chaotic dynamics. Full article

The equilibrium shape of a crystal is a fundamental problem in materials science and condensed matter physics. The Wulff construction, a cornerstone of crystal morphology prediction, is traditionally presented and utilized as a powerful geometric algorithm to derive equilibrium shapes from anisotropic surface energy $\gamma \left(\mathbf{n}\right)$. While its application across materials science is vast, the profound mathematical physics underpinning it, specifically its intrinsic identity as a manifestation of the Legendre transform, is often relegated to a passing remark. This work recenters the focus on this fundamental duality. We present a comprehensive, step-by-step derivation of the Wulff shape from the variational principle of surface energy minimization under a constant volume, employing the language of support functions and differential geometry. We then rigorously demonstrate that the equilibrium shape, defined by the support function $h\left(\mathbf{n}\right)$, and the surface energy density $\gamma \left(\mathbf{n}\right)$ are conjugate variables linked by a Legendre transformation; the Wulff shape $\mathcal{W}$ is precisely the zero-sublevel set of the dual function ${\gamma }^{*}\left(\mathbf{x}\right)={sup}_{\mathbf{n}}[\mathbf{x}·\mathbf{n}-\gamma \left(\mathbf{n}\right)]$. This perspective elevates the Wulff construction from a mere graphical tool to a canonical example of convex duality in thermodynamic systems, connecting it to deeper principles in convex analysis and statistical mechanics. To bridge theory and computation, we provide a robust computational algorithm implemented in pseudocode capable of generating Wulff shapes for two-dimensional (2D) crystals with arbitrary N-fold symmetry. Finally, we discuss the relevance and extensions of the classical theory in contemporary research, including non-equilibrium growth, nanoscale effects, and the coupling of crystal shapes with elastic membrane environments. Full article

In the context of a low-carbon economy, automotive parts supply chains face multifaceted risks, making an effective supply chain risk assessment model a crucial means of ensuring supply chain stability. Traditional evaluation methods struggle to comprehensively and accurately identify all influencing factors and their interrelationships in automotive parts supply chains. This article constructs an evaluation model based on the principle of symmetry. The “structural symmetry” is determined by the ratio of the completeness of risk dimension coverage in the indicator system to the precision of indicators, while “fusion symmetry” refers to the degree of equilibrium in information contribution during the fusion of subjective and objective weights. First, Fault Tree Analysis (FTA) and the Delphi method are adopted to establish a risk evaluation index system, whereby structural symmetry is ensured by the equilibrium between the completeness of risk dimension coverage and the accuracy of indicators in the index system. Second, drawing on the symmetric fusion principle, this study proposes a hybrid evaluation approach integrating hesitant fuzzy DEMATEL with entropy weight-coefficient of variation (HDEC), and the fusion symmetry is guaranteed by the balanced integration of subjective and objective weight information. Finally, a case study of an automotive parts supply chain enterprise quantitatively assesses and ranks risk factors, with corresponding countermeasures proposed. The symmetry-guided HDEC method achieves high accuracy, identifying indicator–causal relationships. Compared with the traditional entropy-weighted AHP algorithm, the Pearson correlation coefficient is 0.8566, and Spearman’s rank correlation coefficient is 0.88, indicating strong weight correlation and robust stability. The integration of mathematical symmetry enhances the model’s theoretical rigor, which aligns with symmetry-oriented optimization research. Full article

Given a network of nodes and a certain demand that needs to be satisfied, the capacitated dispersion problem (CDP) involves selecting a subset of nodes to maximize dispersion between them. In many practical instances, symmetry in the structure of the selected nodes (e.g., using nodes of the same type) can lead to synergies. Hence, this paper studies a bi-objective variant of the CDP to account for these symmetries. The first goal seeks to maximize the minimum distance between opened nodes, while the second goal accounts for symmetry by penalizing the use of nodes of different types (in our case, represented by different colors). We formalize the problem as a bi-objective mathematical program and address it through a classical multiobjective strategy, the $ϵ$-constraint method. Exact methods are used when the problem size allows, even though the problem is NP-hard. To tackle larger instances, we design a biased-randomized algorithm based on a constructive heuristic. Computational experiments show that our biased-randomized algorithm provides high-quality approximations of the Pareto frontier. Full article