Search Results (28)

This paper develops a rigorous mathematical and computational framework for four-dimensional chess defined on the discrete hypercubic lattice $\{1,\dots , 8{\}}^4$. We formalize piece movement using displacement sets in ${\mathbb{Z}}^4$, define adjacency via the Chebyshev metric, and analyze the resulting move graphs for rooks, bishops, knights, queens, and kings. We establish exact mobility formulas, parity invariants, and connectivity properties, consolidating known product-graph results for rooks and kings while introducing a boundary-sensitive analysis of the four-dimensional knight verified by exhaustive enumeration. The mathematical framework is complemented by a fully implemented 4D chess engine and interactive visualization environment rendering all 64 (z,w)-slices of the hypercube simultaneously. The system supports full move legality, generalized special rules, multi-king checkmate detection, and reproducible state enumeration. Performance measurements and exploratory branching-factor estimates are obtained through reproducible random playouts using the publicly available implementation. We contextualize this ruleset within existing work on move graphs on ${\mathbb{Z}}_n^m$, higher-dimensional leapers, spectral properties of grid graphs, toroidal analogs, and multidimensional visualization. Exploratory qualitative feedback (N = 18) is included to examine whether the visualization design is interpretable and navigable in practice, providing feasibility-oriented observations on how slice-based 4D projection and layered board rendering are perceived by non-expert users in an exploratory context. Together, the mathematical results, implemented engine, and visualization form a coherent foundation for the study of strategy, complexity, and human interaction in four-dimensional game systems. The framework provides a basis for future investigations into spectral analysis of move graphs, symmetry-aware search, hierarchical planning, and educational applications in high-dimensional geometry. Full article

Mn-based layered oxide cathodes are pivotal for advancing sodium-ion batteries, yet their practical deployment is hindered by structural instability and complex phase transformations during cycling. This review provides a systematic overview of recent strategies aimed at rational design and performance enhancement of these materials. It begins with fundamental thermodynamic principles governing phase formation, particularly P2/O3 structural dichotomy, and highlights the critical roles of sodium content, transition metal chemistry, and ionic potential in determining crystal stability. The emergence of high-entropy engineering is examined as a powerful approach to suppress detrimental phase transitions through configurational entropy stabilization, lattice distortion, and synergistic multi-element interactions. Furthermore, the integration of machine learning with multidimensional descriptors including electronegativity-weighted entropy and cationic potential enables more accurate predictions of phase behavior in complex compositional spaces. The review also highlights the decisive influence of synthesis protocols, where precise control over calcination conditions, atmosphere, and local elemental distribution enables the formation of targeted phase architectures, such as P2/O3 intergrowth, which exhibit superior electrochemical robustness. Collectively, these advances illustrate a shift from empirical trial and error toward a theory-guided, data-informed framework for designing high-performance layered oxide cathodes. Full article

This paper presents a comprehensive study of nonlinear wave and oscillator dynamics under both deterministic and stochastic influences. By comparing soliton-like and dispersive waveforms, we employ spectral solvers, Darboux transformations, and nonlinear diagnostics, including Lyapunov exponents, power spectral analysis, and multidimensional phase-space reconstructions, to examine transitions from quasiperiodic motion to chaotic and stochastic regimes. The results highlight the robustness of soliton solutions in preserving energy and structure, in contrast to the degradation observed in dispersive waves under noise and damping. We also show that spectral broadening, entropy growth, and ergodic phase-space patterns are caused by the critical influence of initial conditions and noise intensity on system behavior. Incorporating control strategies such as OGY chaos control, this work provides a flexible framework for analyzing, modeling, and stabilizing nonlinear systems. Applications span nonlinear optics, fluid flows, and electrical lattices, offering insight into the interplay of nonlinearity and noise with implications for both theoretical understanding and practical system design. Full article

Shale oil extraction efficiency hinges on the interfacial interactions between oil molecules and reservoir clay minerals, such as kaolinite, whose slit-like pores serve as primary storage spaces for alkanes. This study introduces a novel multi-dimensional quantification method using four angular parameters—elevation angle (θ), azimuth angle (φ), rotation angle (ω), and dihedral angle (τ)—to systematically investigate the adsorption configuration of n-octane in kaolinite slit pores ranging from 0.45 to 14.05 nm. Through molecular simulations and advanced trajectory analysis, we elucidate the impact of pore sizes on alkane adsorption density, layering, and molecular configurations. Results reveal that pore size regulates molecular behavior via steric hindrance and potential field superposition, while the four angular parameters can effectively capture subtle changes in. this molecular behavior: (1) the elevation angle (θ) around 0° indicates complete alignment parallel to surface, but is modulated at increasing distance from the surface into the pore-region highlighting a disordered state; (2) the azimuth angle (φ) is concentrated at 60° and 120° on the siloxane tetrahedral surface due to lattice regulation, but shows a disordered distribution on the hydroxyl octahedral surface; (3) the rotation angle (ω) is mainly concentrated at 0° and 90° indicating molecular plane being either parallel or perpendicular to the surface; (4) the dihedral angle (τ) remains at ~0°, indicating that the molecular chains are straight. In pores smaller than 4.26 nm, strong confinement yields ordered molecular arrangements (θ = 0°, φ at 60° or 120°, ω = 0°) with high adsorption density; for larger pores than 4.26 nm, disordered configurations and increased layering (up to eight layers) with stable density and adsorption capacity per unit area are observed. The proposed parameter system overcomes limitations of traditional qualitative approaches, offering a standardized, scalable tool for quantifying alkane-clay interactions. This framework enhances understanding of shale oil occurrence mechanisms and supports optimized extraction strategies, with broad applicability to other chain molecules and 2D materials in interface science. Full article

In the V2X environment, intelligent connected vehicles can obtain multi-dimensional traffic flow data in real time through the vehicle–road collaborative cyber–physical fusion system. Based on this, this study proposes a multi-lane traffic flow lattice model integrating optimal traffic flow difference estimation information to effectively suppress traffic congestion. The linear stability criterion of the system is derived through linear stability analysis, proving that the optimal traffic flow difference estimation can significantly expand the stable region and suppress traffic fluctuations caused by small disturbances. Furthermore, the perturbation method is used to derive the mKdV equation near the critical stability point of the system, revealing the nonlinear characteristics of traffic congestion propagating in the form of kink solitary waves, and indicating that the new consideration effect can effectively slow down the congestion propagation speed by adjusting the parameters of solitary waves (such as wave speed and amplitude). The numerical simulation results show that compared to the traditional model, the improved model exhibits enhanced traffic flow stability and robustness. Meanwhile, it reveals the nonlinear relationship between the increase of the number of lanes and the alleviation of congestion, and there is an optimal lane configuration threshold. The research results not only provide theoretical support for the optimization of traffic flow efficiency in intelligent transportation systems, but also provide a decision-making basis for dynamic lane management strategies in the V2X environment. Full article

A partially ordered set (or a poset, for short) is a set endowed with a partial order relation, i.e., with a reflexive, anti-symmetric, and transitive binary relation. As mathematical objects, posets have been intensively studied in the last century, coming to play essential roles in pure mathematics, logic, and theoretical computer science. More recently, they have been increasingly employed in data analysis, multi-criteria decision-making, and social sciences, particularly for building synthetic indicators and extracting rankings from multidimensional systems of ordinal data. Posets naturally represent systems and phenomena where some elements can be compared and ordered, while others cannot be and are then incomparable. This makes them a powerful data structure to describe collections of units assessed against multidimensional variable systems, preserving the nuanced and multi-faceted nature of the underlying domains. Moreover, poset theory collects the proper mathematical tools to treat ordinal data, fully respecting their non-numerical nature, and to extract information out of order relations, providing the proper setting for the statistical analysis of multidimensional ordinal data. Currently, their use is expanding both to solve open methodological issues in ordinal data analysis and to address evaluation problems in socio-economic sciences, from multidimensional poverty, well-being, or quality-of-life assessment to the measurement of financial literacy, from the construction of knowledge spaces in mathematical psychology and education theory to the measurement of multidimensional ordinal inequality/polarization. Full article

In this study, a terahertz microfluidic multi-band sensor was designed. Unlike previous microfluidic absorption sensors that rely on dipole resonance, the proposed sensor uses a physical mechanism for absorption by exciting higher-order lattice resonances in microfluidic structures. With a Fabry–Perot cavity, the sensor can form an absorption peak with a high quality factor (Q) and narrow full width at half maximum (FWHM). A high Q value and a narrow FWHM are valuable in the field of sensing and provide strong support for high-precision sensing. On this basis, the sensing performance of the device was investigated. The simulation results clearly show that the absorption sensor has ultra-high sensitivity, which reaches 400 GHz/Refractive Index Unit (RIU). In addition, the sensor generates three absorption peaks, overcoming the limitations of a single frequency band in a composite resonance mode and multidimensional frequency response, which has potential application value in the field of volatile organic compound (VOC) sensing. Full article

Developing high-performance photocatalysts for solar hydrogen production requires the synergistic modulation of chemical composition, nanostructure, and charge carrier transport pathways. Herein, we report a Cs and P co-doped tubular graphitic carbon nitride (Cs/PTCN-x) photocatalyst synthesized via a strategy that integrates elemental doping with morphological engineering. Structural characterizations reveal that phosphorus atoms substitute lattice carbon to form P-N bonds, while Cs⁺ ions intercalate between g-C₃N₄ layers, collectively modulating surface electronic states and enhancing charge transport. Under visible-light irradiation (λ ≥ 400 nm), the optimized Cs/PTCN-3 catalyst achieves an impressive hydrogen evolution rate of 8.085 mmol·g⁻¹·h⁻¹—over 33 times higher than that of pristine g-C₃N₄. This remarkable performance is attributed to the multidimensional synergy between band structure tailoring and hierarchical porous tubular architecture, which together enhance light absorption, charge separation, and surface reaction kinetics. This work offers a versatile approach for the rational design of g-C₃N₄-based photocatalysts toward efficient solar-to-hydrogen energy conversion. Full article

To address the growing threat of quantum computing to classical cryptographic primitives, this study introduces NTRU-MCF, a novel lattice-based signature scheme that integrates multidimensional lattice structures with fractional-order chaotic systems. By extending the NTRU framework to multidimensional polynomial rings, NTRU-MCF exponentially expands the private key search space, achieving a key space size $\ge 2^{256}$ for dimensions $m\ge 2$ and rendering brute-force attacks infeasible. By incorporating fractional-order chaotic masks generated via a hyperchaotic Lü system, the scheme introduces nonlinear randomness and robust resistance to physical attacks. Fractional-order chaotic masks, generated via a hyperchaotic Lü system validated through NIST SP 800-22 randomness tests, replace conventional pseudorandom number generators (PRNGs). The sensitivity to initial conditions ensures cryptographic unpredictability, while the use of a fractional-order L hyperchaotic system—instead of conventional pseudorandom number generators (PRNGs)—leverages multiple Lyapunov exponents and initial value sensitivity to embed physically unclonable properties into key generation, effectively mitigating side-channel analysis. Theoretical analysis shows that NTRU-MCF’s security reduces to the Ring Learning with Errors (RLWE) problem, offering superior quantum resistance compared to existing NTRU variants. While its computational and storage complexity suits high-security applications like military and financial systems, it is less suitable for resource-constrained devices. NTRU-MCF provides robust quantum resistance and side-channel defense, advancing PQC for classical computing environments. Full article

The increasing frequency of urban flooding, driven by global climate change, poses significant threats to the safety and resilience of urban rail transit systems. This study systematically examines the cascading failure processes and resilience of these networks under flood conditions, with a specific focus on the Shanghai Metro. A comprehensive resilience evaluation model was developed by integrating geographic information, static network characteristics, and dynamic passenger flow indicators. This study employs an improved Coupled Map Lattice (CML) model to simulate cascading failures by considering the coupling effects of station centrality, geographic elevation, and passenger flow dynamics. The results indicate that stations with higher degrees of centrality are more likely to trigger rapid cascading failures across the network. However, incorporating dynamic passenger flow and geographic elevation data helps mitigate these effects, emphasizing the need for multi-dimensional resilience strategies. The findings provide valuable insights for urban transit management, offering a scientific foundation for developing targeted disaster response strategies to enhance network resilience against floods. This study advances our understanding of the vulnerability of urban rail transit systems and offers practical guidance for improving disaster preparedness in urban transportation infrastructure. Full article

The fractionation in microchannels is a promising approach for the delivery of microparticles in narrow property distributions. The underlying mechanisms of the channels are however often not completely understood and are therefore subject to current research. These investigations are done using different numerical and experimental methods. In this work, we present and evaluate our method of combining a numerical Discrete Element Method (DEM)-Lattice Boltzmann Method (LBM) approach with experimental long-exposure fluorescence microscopy, micro-Particle Image Velocimetry (µPIV) and Astigmatism Particle Tracking Velocimetry (APTV) measurements. The suitability of the single approaches and their synergies are evaluated using the exemplary investigation of multidimensional fractionation in different channel geometries. It shows that both, numerical and experimental method are well suited to evaluate particle dynamics in microchannels. As they furthermore show strengths canceling out weaknesses of the respective other method, the combined method is very well suited for the comprehensive analysis of particle dynamics in microchannels. Full article

This work addresses the effects of blade tip-rake reformation on the performance of marine propellers using a low-cost potential-based vortex-lattice method (VLM) and the high fidelity artificial compressibility CFD-RANS solver MaPFlow. The primary focus lies on determining whether the low-cost VLM, in conjunction with a multidimensional parametric model for the tip-rake and pitch/camber distributions, can produce a propeller geometry with improved efficiency. Due to the availability of experimental and numerical data, the NSRDC 4381-82 propellers were selected as reference geometries. Torque minimization serves as the objective function in the gradient-based optimization procedure under a thrust constraint, which translates into efficiency enhancement at the selected design advance ratio. The optimized 4381 propeller yields a +1.1% improvement in efficiency based on CFD-RANS, whereas for the modified skewed 4382 propeller, the efficiency gain is +0.5%. The performance enhancement is also evident at a region near the design advance ratio. The results suggest that the exploitation of low-cost VLM solvers can significantly reduce the CFD simulations required in the optimization process and thus can be effectively used for the design of propellers with tip-rake reformation. Full article

With changes in the city environment and advances in engineering technologies, there is an increasing demand for the construction of super-large span city domes that can cover a large area to create a small internal environment within a specific region. However, the structural design must overcome various challenges in order to break the current structural span limitations. Moreover, there is little research on structures achieving such large spans. The seismic performance of the selected Kiewitt-type, Geodesic-type, and Three-dimensional grid-type mega-latticed structures is further investigated upon previous studies of the model selection, static and stability analysis results of the 800 m span mega-latticed structures. Finite element models were established with ANSYS to analyze the modal properties and earthquake response of the structures. The study evaluated the impact of earthquake directionality on the structural response as well as the response pattern of the structure under frequent and rare earthquake actions. It was found that the overall integrity of the structures is good, with strong coupling effects in three directions. The multi-dimensional seismic input method should be applied to solve the structural response. Combining the plastic development of the structure under rare earthquakes, the top and the circumferential trusses of the third and fourth rings are relatively weak parts of the structures. According to this study, given the known static analysis results, the maximum displacement and maximum stress of the structures under frequent and rare earthquake actions can be estimated. Furthermore, the study highlights that Three-dimensional grid-type mega-latticed structures should be prioritized designing structures with spans of 800 m, providing helpful guidance for the practical application of this type of structure. Full article

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point. Full article

An infrared photodetector is a critical component that detects, identifies, and tracks complex targets in a detection system. Infrared photodetectors based on 3D bulk materials are widely applied in national defense, military, communications, and astronomy fields. The complex application environment requires higher performance and multi-dimensional capability. The emergence of 2D materials has brought new possibilities to develop next-generation infrared detectors. However, the inherent thickness limitations and the immature preparation of 2D materials still lead to low quantum efficiency and slow response speeds. This review summarizes 2D/3D hybrid van der Waals heterojunctions for infrared photodetection. First, the physical properties of 2D and 3D materials related to detection capability, including thickness, band gap, absorption band, quantum efficiency, and carrier mobility, are summarized. Then, the primary research progress of 2D/3D infrared detectors is reviewed from performance improvement (broadband, high-responsivity, fast response) and new functional devices (two-color detectors, polarization detectors). Importantly, combining low-doped 3D and flexible 2D materials can effectively improve the responsivity and detection speed due to a significant depletion region width. Furthermore, combining the anisotropic 2D lattice structure and high absorbance of 3D materials provides a new strategy in high-performance polarization detectors. This paper offers prospects for developing 2D/3D high-performance infrared detection technology. Full article