Search Results (155)

We formulate a gauge-invariant phenomenological theory for a charged three-component condensate in an anisotropic layered superconductor. The construction is conditional on a material-specific pairing calculation selecting an isolated, predominantly triplet channel; it does not infer triplet pairing from layering or spin–orbit coupling alone. The order parameter is represented equivalently by a spin-1 spinor, a complex d-vector, and a pure-vector complex quaternion. Only the mapping and the density-spin bilinear are retained in the main text. A static Ginzburg–Landau functional then yields axial-polar, planar-polar, easy-axis polarized, and broken-axisymmetry mean-field states. Conservative Gross–Pitaevskii dynamics are introduced only as an additional composite-boson limit, not as a generic consequence of the Ginzburg–Landau theory. In that limit, analytic spectra are given for the axial-polar and easy-axis phases: the transverse spin branch softens at the axial-polar to broken-axisymmetry boundary, whereas crystal locking gaps the transverse magnon of the polarized phase. We do not claim a closed analytic spectrum for the mixed broken-axisymmetry phase. The static transverse current-response kernel provides a quantitative link between penetration-depth anisotropy and the gradient tensor. Ideal Bose condensation and BKT formulas are stated only in their controlled three- and two-dimensional limits. The framework therefore supplies a compact, falsifiable set of phase, mode, and response relations without introducing additional quaternionic degrees of freedom. Full article

Quaternion-based moments and transforms have emerged as powerful tools for color image processing, attracting substantial research interest. Polar harmonic transforms (PHTs) have been shown to be a class of continuous orthogonal moments with good performance in image representation and reconstruction. In this paper, we propose a novel quaternion polar transform, termed the generalized quaternion polar linear canonical transform (GQPLCT). The proposed transform employs a six-parameter kernel, offering both flexibility and generality. By assigning special values to the parameters, it can degenerate into several existing quaternion polar harmonic transforms. Furthermore, we analyze its scaling invariance and establish its relationship to the complex GPLCT to enable efficient computation. Experimental results demonstrate the effectiveness of the proposed method and its favorable performance in terms of scale invariance, numerical stability, and image representation. Full article

Multilayer perceptrons (MLPs) are considered as a singular model of learning machines. Singularities cause local minima and plateaus in the learning process. I/O-equivalence, where two different MLPs are regarded as the same multivariable function, is an important concept for understanding singularities in neural networks. In this paper, I/O-equivalence is extended to T-equivalence, which is a concept where two MLPs yield the same results through a transformation of input and output. We provide constructive families and procedures for obtaining T-equivalent networks of real-, complex-, and quaternion-valued neural networks. In particular, T-equivalence of quaternion-valued neural networks is much more complicated than that of the others. Full article

The paper proposes two attitude motion determination algorithms based on an extended Kalman filter dealing with asynchronous measurements of the star tracker and angular velocity sensor. The satellite control system tracks a complex angular velocity profile during the Earth-point stabilization of the remote sensing camera axis. The star tracker measurement accuracy decreases with higher satellite angular velocity; when a certain angular velocity value threshold is exceeded, the star tracker measurements might be unavailable. These star tracker aspects, along with the variable bias of the angular velocity sensor, are addressed by the developed algorithms. The first algorithm uses kinematic relations; it estimates the attitude quaternion and the angular velocity sensor bias. It does not require information on satellite inertia parameters and on current control torque; it is characterized by low computational burden, though the angular velocity estimation accuracy is limited by the standard deviation of the sensor random noise. The second algorithm is based on both kinematic and dynamic motion equations: it estimates the attitude quaternion, angular velocity sensor bias, and angular velocity vector as well. The satellite tensor of inertia, reaction wheels’ parameters, and history of control inputs are required for the state vector estimation. The performance of these algorithms is compared under the scenario of Earth-point stabilization attitude motion, taking into account different levels of angular velocity measurement errors. It is obtained that the algorithm based on kinematic equations only is characterized by lower stabilization and estimation accuracies compared to the algorithm based on both kinematic and dynamic motion equations, though the latter is significantly more computationally complex. The influence of the state vector estimation errors on the Earth-point stabilization accuracy is studied. The paper contribution is an algorithm performance study considering star-tracker accuracy degradation with angular velocity during the Earth-point flyby. Full article

Sixth-generation (6G) networks require advanced non-orthogonal multiple access (NOMA) schemes to support the technical requirements of dense massive machine-type communications (mMTC). While sparse-code multiple access (SCMA) improves uplink spectral efficiency, its operation within the complex two-dimensional domain ($\mathbb{C}$) generates spatial congestion and high error rates under high-load scenarios. Furthermore, when using dual-polarization transceivers, cross-polarization discrimination leakage is not efficiently exploited because it operates in a conventional 2D environment. This work proposes a hypercomplex transmission architecture, called quaternionic SVD-SCMA (Q-SVD-SCMA), which maps SCMA codewords to a quaternionic group ($Q_8$) in ${\mathbb{R}}^4$. The proposed scheme uses purely imaginary spatial rotators to project overlapping signals onto mutually orthogonal geometric subspaces, thus mitigating interference between users. On the receiver side, a quaternionic sphere decoder (Q-SD) is proposed to evaluate the minimum quaternionic Euclidean distance (MQED) to provide near-optimal detection. Computational simulations are performed under a doubly polarized Rayleigh fading channel with energy normalization to decouple arbitrary power-scale topological gains. To evaluate system performance, both the symbol error rate (SER) and the bit error rate (BER) are used. The results obtained demonstrate that Q-SVD-SCMA effectively transforms Cross-Polarization Discrimination (XPD) leakage into spatial diversity gain. The hypercomplex 4D architecture proposed in this work eliminates the interference-induced error threshold and limits the bit error penalty through multidimensional Gray mapping, providing a scalable and highly reliable physical layer framework for overloaded 6G scenarios, unlike its conventional 2D predecessors (C-SCMA and SVD-SCMA). Full article

Accurate reconstruction of complex pavement surface textures is essential for reliable texture characterization and skid resistance evaluation. In this study, a three-dimensional point cloud correction method based on a quaternion rotation matrix was proposed to eliminate point cloud tilting and displacement caused by pavement slope and the initial orientation of the scanning equipment. A unified spatial rotation model was established to align the point cloud normal vector with the Z-axis while preserving the geometric relationships among points. The method was validated using a pavement model with regular sharp textures and then applied to distressed pavements containing cracks, raveling, and rutting. The results showed that the proposed method more accurately reconstructed the original elevation features and effectively avoided the peak–valley blunting and texture smoothing associated with conventional profile-fitting correction. Compared with the conventional method, the average error of mean texture depth (MTD) was reduced by 48.3%, while the error of maximum texture elevation difference (Δh) remained within 39.9%. Statistical analysis indicated that the observed differences mainly resulted from the correction strategy rather than measurement uncertainty. The proposed method improves the accuracy of pavement texture reconstruction and texture parameter calculation and provides a reference for pavement evaluation and digital pavement modeling. Full article

Algebraic methodologies are developed for addressing the inverse problem of recovering a collection of bivariate observations from selected subsets of their marginal and joint integer moments. Particular attention is devoted to the case of samples containing duplicates. The multivariate analogues emerge from suitable generalizations of the bivariate constructions. We also examine the problem of reconstructing a sample from its marginal moments together with its component-wise ranks. Extensions to complex-valued data are also considered. The results established in this paper demonstrate that an appropriately chosen family of marginal or joint moments can, in fact, encode the entirety of the information contained in the original data set, in the sense that the latter can be recovered uniquely from the former. The proposed methodologies are essentially mathematical, with statistical samples representing one of several possible applications. A series of illustrative examples highlights the scope of the methods. Full article

This paper develops a direct analytical framework for synchronizing and controlling fractional-order octonion-valued fuzzy bidirectional associative memory neural networks (FOOVFBAMNNs). Octonion algebra is neither commutative nor associative, which limits the application of standard analytical tools. To address this challenge, we first propose a generalized Cauchy–Schwarz inequality tailored to the octonionic domain, which operates directly without relying on system decomposition. This inequality lays the groundwork for a Lyapunov-based stability analysis that retains the system’s inherent geometric structure to avoid decomposition into real-valued components. Based on this framework, we derive concise 2-norm inequality criteria, which are sufficient to guarantee Mittag-Leffler synchronization of the proposed model. We also employ a Particle Swarm Optimization (PSO) algorithm to systematically optimize the flexible parameters in the generalized inequality, enhancing the practical performance of the synchronization scheme. To validate the effectiveness of the proposed method, we apply it to a multi-domain image restoration task. Numerical experiments verify the performance of our method. In terms of Peak Signal-to-Noise Ratio (PSNR), the octonion-valued network with PSO-tuned parameters achieves better results than its non-optimized counterpart as well as models constructed in complex or quaternion domains. Full article

We develop a comparative synthesis of quantum-information geometry beyond complex numbers, with emphasis on what different algebraic frameworks contribute to information-processing structure rather than on their formal novelty alone. The organizing idea is a layer-by-layer test of the standard complex Hilbert-space formalism: each non-complex or deformed framework modifies the scalar field, phase group, projective state space, Born-probability semantics, composition rule, measurement geometry, symmetry algebra or representation category. The central thesis is that such frameworks are physically meaningful when they identify which assumptions make complex quantum mechanics operationally stable: positive probabilities, associative multipartite composition, reversible dynamics, experimentally testable phases, locality constraints, informationally complete measurements, error bases and clear operational semantics. Real quantum theory probes the necessity of complex phases and local tomography; quaternionic quantum mechanics probes non-Abelian phase while retaining associativity and admitting complex embeddings; octonionic proposals probe the boundary where exceptional geometry survives but generic circuit composition is obstructed by non-associativity; Jordan algebras test ordered probabilistic state spaces; Clifford algebras and Bott periodicity provide the spinorial and topological grammar connecting gates, Hopf maps and periodic dimensions; and quantum-group or q-deformed constructions probe coproducts, braiding and representation categories rather than scalar amplitudes. We distinguish three roles that are often conflated: genuine hypercomplex kinematics, Hopf-fibration coordinates for ordinary complex multipartite entanglement, and deformed algebraic or categorical structures. The resulting map separates established equivalence and experimental-constraint results from useful representation tools and speculative programs, while identifying concrete open problems for non-complex quantum information. Full article

We construct solutions of a time-harmonic Maxwell-type system within the framework of the algebra of complex quaternions. Using quaternionic analysis, we establish a connection between this system and certain first-order differential operators whose kernels consist of monogenic functions. Building on known representations of harmonic and monogenic functions, we develop a constructive procedure based on transmutation operators for generating explicit solutions of the equations $(D\pm \lambda )u=0$, and consequently of the corresponding Maxwell system. This approach provides a systematic method for reconstructing electromagnetic fields from harmonic and monogenic data, yielding an explicit link between quaternionic operator theory, transmutation methods, and the classical formulation of Maxwell equations. Full article

Dynamic Movement Primitives (DMPs) are widely used for learning and generalizing robot skills. However, standard quaternion DMPs, when modeling orientation trajectories, constrain only the final orientation and cannot freely specify the final angular velocity. This limitation restricts its application to dynamic tasks requiring precise boundary conditions, such as hitting or throwing. Although existing improved methods achieve velocity generalization to some extent, they often struggle to balance trajectory shape preservation with dynamic smoothness, frequently causing significant deviation from demonstrations or abrupt acceleration discontinuities. In this paper, we propose a novel robot skill generalization method that enables controllable final angular velocity for quaternion DMPs. Specifically, we construct a dynamic goal system driven by a quintic polynomial in Lie algebra space, analytically planning the target orientation’s evolution based on given multi-order boundary constraints. This mechanism not only achieves precise control over the final angular velocity but also inherently guarantees global C² continuous dynamics across primitive segments. Comparative simulations and real-world robot hitting experiments demonstrate that, compared to existing approaches, our proposed method effectively satisfies dynamic boundary constraints while exhibiting superior shape preservation, minimal trajectory deviation, and higher smoothness, thereby significantly improving skill generalization performance in complex dynamic tasks. Full article

Complex metrics play a fundamental role in applied mathematical analysis and in the study of physical phenomena. In this work, two complex Minkowski spacetime metrics are examined in detail. The investigation centers on the spin matrices that generate these spacetimes, their corresponding algebraic properties, and the quaternionic structures that emerge from them. The key findings of this study include the identification of a novel set of spin matrices and the characterization of their symmetry, spectral, commutator, and anticommutator properties. Furthermore, a new generalized Lorentz algebra is derived, and a quaternionic mapping of the proposed spin matrices is performed, leading to the construction of new orthonormal quaternionic basis vectors. Full article

In precision agriculture, identifying pests such as the whitefly (Bemisia tabaci) is a significant challenge, as precise knowledge of these insects is essential for developing effective Integrated Pest Management (IPM) strategies. Automated daily monitoring within IPM programs optimizes the diagnostic registration stage by reducing logistical expenses and manual errors, enabling early pest treatment interventions and providing quantitative data for informed decision-making. In this study, an image bank was processed using a Quaternionic Gabor Filter (QGF) algorithmto highlight textural features through hypercomplex correlation. The highlighted objects were then processed by a YOLOv8 pretrained model to identify Bemisia tabaci. Experimental results demonstrate that this combination achieves a precision of 0.868 and an mAP@0.5 of 0.950, while a PSNR of 34.10 dB ensures the structural integrity of the enhanced images. Although the total execution time averages 2.3 s per image due to preprocessing complexity, the GPU inference time of 10.3 ms confirms the potential for high-speed detection. This approach significantly enhanced the morphological features of Bemisia tabaci, increasing the robustness of the detection model and narrowing down processing conditions for yellow trap samples to strengthen precision in the semi-arid regions of Zacatecas, Mexico. Full article

This paper proposes a novel attitude control methodology for unmanned aerial vehicles (UAVs). The core of the approach is a decomposition of the vehicle’s orientation into a distinct horizontal tilt component and a yaw rotation. This reformulation provides a principled theoretical framework by redefining attitude error within an acceleration-coefficient space, which linearizes and decouples the horizontal acceleration dynamics—a significant simplification over conventional Euler-angle or quaternion-based models. From an engineering perspective, this framework enables the direct generation of angular velocity commands that guarantee first-order tracking of desired acceleration coefficients. This leads to a substantial reduction in the computational complexity of the outer-loop controller, enhances trajectory tracking accuracy, and ensures predictable performance under given attitude constraints and angular velocity limits. Simulation results demonstrate the method’s superior performance compared to standard techniques, validating its dual value as both a theoretically-grounded control design tool and an effective solution for high-precision UAV navigation in real-world applications. Full article

Traditional pedestrian inertial navigation (PDR) algorithms usually assume that the carrying mode of a smartphone is fixed and remains horizontal, while ignoring the significant impact of dynamic changes in the carrying mode on heading estimation, which is the core element of PDR algorithms. In practical application scenarios, pedestrians often change their way of carrying smart terminals (e.g., calling) according to their needs, corresponding to the difference in the heading estimation method; especially when the mode is switched, it will cause a sudden change in heading, which will lead to a significant increase in the localization error if it cannot be corrected in time. Existing smart terminal carrying mode recognition methods that rely on traditional machine learning or set thresholds have poor robustness; lack of universality, especially weak diagnostic ability for mutation; and can not effectively reduce the heading error. Based on these practical problems, this paper innovatively proposes a PDR framework that tries to overcome these limitations. Based on this research purpose, firstly, this paper classifies four types of common carrying modes based on practical applications and designs a CNN-LSTM hybrid model, which can classify the four common carrying modes in near real-time, with a recognition accuracy as high as 99.68%. Secondly, based on the mode recognition results, a multi-layer heading correction strategy is introduced: (1) introducing a quaternion-based universal filter (VQF) algorithm to realize the accurate estimation of initial heading; (2) designing an algorithm to accurately detect the mode switching point and developing an adaptive offset correction algorithm to realize the dynamic compensation of heading in the process of mode switching to reduce the impact of sudden changes; and (3) considering the motion characteristics of pedestrians walking in a straight line segment where lateral displacement tends to be close to zero. This study designs a heading optimization method with lateral displacement constraints to further inhibit the drifting of the heading caused by the slight swaying of the smart terminal. In this study, two validation experiments are carried out in two different environment—an indoor corridor and a tree shelter—and the results show that based on the proposed multi-layer heading optimization strategy, the average heading error of the system is lower than 1.5°, the cumulative positioning error is lower than 1% of the walking distance, and the root mean square error of the checkpoints is lower than 2 m, which significantly reduces the positioning error and shows the effectiveness of the framework in complex environments. Full article