Search Results (77)

We consider the nonlinear reaction–diffusion equation on the unit sphere $u_t={\mathrm{\Delta }}_{{\mathbb{S}}^2}u+f\left(u\right)$, $f_{uu}\ne 0$, and carry out a complete Lie point symmetry analysis. Solving the associated determining system yields a rigidity theorem: for every genuinely nonlinear $f\left(u\right)$, the admitted symmetry algebra is $\mathfrak{so}\left(3\right)\oplus ⟨{\partial }_t⟩$, generated by the rotational Killing fields and time translation. We further show through a group classification that the source families that enlarge symmetries in Euclidean space do not produce any additional point symmetries on ${\mathbb{S}}^2$. From an optimal system of subalgebras, we derive curvature-adapted reductions in which the Laplace–Beltrami operator becomes a Legendre-type operator in intrinsic invariants. For the specific nonlinear source $f\left(u\right)=-e^u-2$, specific reduced ODEs admit a hidden one-parameter symmetry, yielding a first integral and explicit steady states on ${\mathbb{S}}^2$. Full article

Stable attitude control of unmanned or autonomous operations of vehicles moving in three spatial dimensions is essential for safe and reliable operations. Rigid body attitude control is inherently a nonlinear control problem, as the Lie group of rigid body rotations is a compact manifold and not a linear (vector) space. Prior research has shown that the largest possible domain of convergence is provided by smooth attitude feedback control laws are obtained using a Morse function on $\mathrm{SO}\left(3\right)$ as a measure of the attitude stabilization or tracking error. A polar Morse function on $\mathrm{SO}\left(3\right)$ has four critical points, which precludes the possibility of global convergence of the attitude state. When used as part of a Lyapunov function on the state space (the tangent bundle $\mathrm{TSO}\left(3\right)$) of attitude and angular velocity, it gives a globally continuous state-dependent feedback control scheme with the minimum of the Morse function as the almost globally asymptotically stable (AGAS) attitude state. In this work, we explore the use of explicitly time-varying gains for Morse functions for rigid body attitude control. This strategy leads to discrete switching of the indices of the three non-minimum critical points that correspond to the unstable equilibria of the feedback system. The resulting time-varying feedback controller is proved to be AGAS, with the additional desirable property that the time-varying gains destabilize the (locally) stable manifolds of these unstable equilibria. Numerical simulations of the feedback system with appropriate time-varying gains show that a trajectory starting from an initial state close to the stable manifold of an unstable equilibrium, converges to the desired stable equilibrium faster than the corresponding feedback system with constant gains. Full article

Engaging in unethical behaviours, such as cheating, lying, stealing and fraud, holds significant consequences for individuals and the broader community. Drawing on self-determination theory, we posit that in a consumer-centric society, where one’s worth is often linked to wealth, celebrity status, and appearance, individuals who adopt extrinsic values might be motivated to engage in unethical behaviour in pursuit of financial gains. Study 1 surveyed university students in Macao, China (n = 566), and crowdsourcing workers from the UK (n = 605), demonstrating that extrinsic values were linked to unethical decision-making in vignette-based scenarios. This association was held in both societies, suggesting a culture-independent connection between unethicality and values. To establish causal relationships, we conducted experiments manipulating extrinsic cues participants received in Macanese (Study 2, n = 170) and British (Study 3, n = 197) participants. Results revealed a significant impact of these cues on behavioural measures of unethicality, with those in the extrinsic-cues condition more likely to lie for financial gains compared to those in the control groups. Together, these findings highlight the influence of extrinsic values on unethical behaviour across cultural contexts. This research underscores the urgent need to address societal norms and consumerist cues that focus on extrinsic values, which may erode ethical standards and threaten collective well-being. Full article

In this paper, the use of nilpotent Lie algebras as the basis for homomorphic encryption based on additive operations is explored. The $g$-setting is set up over $g{l}_n\left({\mathbb{Z}}_q\right)$) and the group $G=exp\left(g\right)$, and it is noted that the exponential and logarithm series are truncated by nilpotency in a natural way. From this, an additive symmetric conjugation scheme is constructed: given a message element $M$ and a central randomizer $U\in z\left(g\right),$ we encrypt $=Kexp\left(M+U\right)K^{-1}$ and decrypt to $M=log\left(K^{-1}CK\right)-U$. The scheme is additive in nature, with the security defined in the IND-CPA model. Integrity is ensured using an encrypt-then-MAC construction. These properties together provide both confidentiality and robustness while preserving the homomorphic functionality. The scheme realizes additive homomorphism through a truncated BCH-sum, so it is suitable for ciphertext summations. We implemented a prototype and took reproducible measurements (Python 3.11/NumPy) of the series $\{{10,10}^2,{10}^3,{10}^4,{10}^5\}$ over 10 iterations, reporting the medians and 95% confidence intervals. The graphs exhibit that the latency per operation remains constant at fixed values, and the total time scales approximately linearly with the batch size; we also report the throughput, peak memory usage, $\mid C\mid /\mid M\mid$ expansion rate, and achievable aggregation depth. The applications are federated reporting, IoT telemetry, and privacy-preserving aggregations in DBMS; the limitations include its additive nature (lacking general multiplicative homomorphism), IND-CPA (but not CCA), and side-channel resistance requirements. We place our approach in contrast to the standard FHE building blocks BFV/BGV/CKKS nd the emerging NIST PQC standards (FIPS 203/204/205), as a well-established security model with future engineering optimizations. Full article

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

The Lorentz group ${\mathrm{Lor}}_{1,3}={\mathrm{SO}}_0(1,3)$ has two point fixgroups, namely $\mathrm{SO}\left(3\right)$ for time-like translations and ${\mathrm{SO}}_0(1,1)\times {\mathrm{I}\mathrm{R}}^2$ for light-like translations. However, for light-like translations, it is reasonable to consider a line fixgroup that leads to the Borel structure of the Lorentz group and provides appropriate helicities for massless particles. Therefore, whether a particle is massless or massive is not so much a physical question but rather a question of the underlying Lie group symmetry. Full article

The convective Cahn–Hilliard–Oono equation is analyzed under the conditions ${\mu }_1\ge 0$ and ${\mu }_3+{\mu }_4\le 0$. The Lie invariance criteria are examined through symmetry generators, leading to the identification of Lie algebra, where translation symmetries exist in both space and time variables. By employing Lie group methods, the equation is transformed into a system of highly nonlinear ordinary differential equations using appropriate similarity transformations. The extended direct algebraic method are utilized to derive various soliton solutions, including kink, anti-kink, singular soliton, bright, dark, periodic, mixed periodic, mixed trigonometric, trigonometric, peakon soliton, anti-peaked with decay, shock, mixed shock-singular, mixed singular, complex solitary shock, singular, and shock wave solutions. The characteristics of selected solutions are illustrated in 3D, 2D, and contour plots for specific wave number effects. Additionally, the model’s stability is examined. These results contribute to advancing research by deepening the understanding of nonlinear wave structures and broadening the scope of knowledge in the field. Full article

This study addresses the nonlinear estimation problem in the strapdown inertial navigation system (SINS) and Doppler velocity log (DVL) integrated navigation by proposing an improved filtering algorithm based on $S{E}_2(3)$ Lie group state representation. A dynamic model satisfying the group affine condition is established to systematically construct both left-invariant and right-invariant error state spaces, upon which two nonlinear filtering approaches are developed. Although the fast unscented transformation method is not novel by itself, its first integration with the $S{E}_2(3)$ Lie group model for SINS/DVL integrated navigation represents a significant advancement. Experimental results demonstrate that under large misalignment angles, the proposed method achieves slightly lower attitude errors compared to linear approaches, while also reducing position estimation errors during dynamic maneuvers. The 12,000 s endurance test confirms the algorithm’s stable long-term performance. Compared with conventional unscented Kalman filter methods, the proposed approach not only reduces computation time by 90% but also achieves real-time processing capability on embedded platforms through optimized sampling strategies and hierarchical state propagation mechanisms. These innovations provide an underwater navigation solution that combines theoretical rigor with engineering practicality, effectively overcoming the computational efficiency and dynamic adaptability limitations of traditional nonlinear filtering methods. Full article

Privacy-sensitive electricity scene classification requires robust models under data localization constraints, making federated learning (FL) a suitable framework. Existing FL frameworks face two critical challenges in multi-label electricity scene classification: (1) Label correlations and their strengths significantly impact classification performance. (2) Electricity scene data and labels show distributional inconsistencies across regions. However, current FL frameworks lack explicit modeling of label correlation strengths, and locally trained regional models naturally capture these differences, leading to regional differences in their model parameters. In this scenario, the server’s standard single-stage aggregation often over-averages the global model’s parameters, reducing its discriminative ability. To address these issues, we propose FMMAN, a federated multi-stage attention neural network for multi-label electricity scene classification. The main contributions of this FMMAN lie in label correlation learning and the stepwise model aggregation. It splits the client–server interaction into multiple stages: (1) Clients train models locally to encode features and label correlation strengths after receiving the server’s initial model. (2) The server clusters these locally trained models into K groups to ensure that models within a group have more consistent parameters and generates K prototype models via intra-group aggregation to reduce over-averaging. The K models are then distributed back to the clients. (3) Clients refine their models using the K prototypes with contrastive group-specific consistency regularization to further mitigate over-averaging, and sends the refined model back to the server. (4) Finally, the server aggregates the models into a global model. Experiments on multi-label benchmarks verify that FMMAN outperforms baseline methods. Full article

Negative-pressure wound therapy (NPWT) using vacuum-assisted closure (VAC) is a well known tissue defect bridging method that applies a vacuum pump to sterile, open-cell foam dressings via suction tubes. Although it has mostly been described for soft tissue use, there are also a few studies concerning its use on hard tissue. However, as oral and maxillofacial surgery has to deal with both soft and hard tissue, which lie next to each other in these regions, there is a particular need to assess the influence of negative pressure on bone. Therefore, the effects of different negative pressure levels (530 mbar and 725 mbar) and atmospheric pressure (1013 mbar) on bone tissue cultures and osteoblast cell cultures were investigated over periods of 1, 3, and 6 weeks. During the culture period, osteoblast growth and the tissue regeneration of bone defects were studied in vitro using tissue cultures that were histologically supplemented by cytological investigations and quantitative RNA expression studies. In the bone defect model, there was a faster defect reduction using NPWT; the effect was especially strong for 530 mbar. Compared to the control group, up to 30% more newly generated bone tissue was detected. This effect on the mineralization capacity was assessed by the mRNA expression of osteogenic marker genes, as well as the receptor activator of nuclear factor κB ligand (RANKL) and osteoprotegerin (OPG), two multifaceted cytokines that regulate bone metabolism. The influence of negative pressure consequently resulted in a decreased RANKL/OPG ratio in osteoblasts. Associated with the upregulation of marker genes to up to 400%, including Col1, BMP4, OCN, and RUNX2, the decrease in the RANKL/OPG ratio to 41% indicates the stimulation of osteogenesis. Since VAC has been shown to be a safe and effective method to close wounds in general, these data suggest that patients suffering from compound bone and soft tissue defects in the maxillofacial area may benefit from an adapted therapy approach accelerating both soft and hard tissue regeneration. Full article

Let X be a compact Riemann surface of genus $g\ge 2$ and $M\left(G_2\right)$ be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type $G_2$. In this work, it is proved that the only automorphisms that $M\left(G_2\right)$ admits are those defined as the pull-back action of an automorphism of the base curve X. The strategy followed uses specific techniques that arise from the geometry of the gauge group $G_2$. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of $G_2$-bundles over X have been proved in the paper. The inclusion of groups $G_2\hookrightarrow Spin(8,\mathbb{C})$ where $G_2$ is viewed as the fixed point subgroup of an order of 3 automorphisms of $Spin(8,\mathbb{C})$ that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles $M\left(G_2\right)\to M(Spin(8,\mathbb{C}))$. In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of $M\left(G_2\right)$ in relation to $M(Spin(8,\mathbb{C}\left)\right)$. Full article

We fit the three finestructure constants of the Standard Model, in which the first approximation of theoretically estimable parameters include (1) a “unified scale”, turning out not equal to the Planck scale and thus only estimable by a very speculative story, the second includes (2) a “number of layers” being a priori the number of families, and the third is (3) a unified coupling related to a critical coupling on a lattice. So formally, we postdict the three fine structure constants! In the philosophy of our model, there is a physical lattice theory with link variables taking values in a (or in the various) “small” representation(s) of the standard model Group. We argue for that these representations function in the first approximation based on the theory of a genuine $SU\left(5\right)$ theory. Next, we take into account fluctuation of the gauge fields in the lattice and obtain a correction to the a priori $SU\left(5\right)$ approximation, because of course the link fluctuations not corresponding to any standard model Lie algebra, but only to the SU(5), do not exist. The model is a development of our old anti-grand-unification model having as its genuine gauge group, close to fundamental scale, a cross-product of the standard model group $S\left(U\right(3)\times U(2\left)\right)$ with itself, with there being one Cartesian product factor for each family. In our old works, we included the hypothesis of the “multiple point criticallity principle”, which here effectively means the coupling constants are critical on the lattice. Counted relative to the Higgs scale, we suggest in our sense that the“unified scale” (where the deviations between the inverse fine structure constants deviate by quantum fluctuations being only from standard model groups, not SU(5)) makes up the 2/3rd power of the Planck scale relative to the Higgs scale or the topquarkmass scale. Full article

In this paper, we investigate the algebraic conditions of algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Furthermore, we provide the complete classification for these algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the algebraic Schouten solitons. The main results indicate that $G_4$ does not possess algebraic Schouten solitons related to the perturbed Kobayashi–Nomizu connection, $G_1,G_2,G_3,G_6$, and $G_7$ possess algebraic Schouten solitons, and the result for $G_5$ is trivial. Full article

In this paper, we address the problem of estimating the position of a mobile such as a drone from noisy position measurements using the framework of Lie groups. To model the motion of a rigid body, the relevant Lie group happens to be the Special Euclidean group $SE\left(n\right)$, with $n=2$ or 3. Our work was carried out using a previously used parametric framework which derived equations for geodesic regression and polynomial regression on Riemannian manifolds. Based on this approach, our goal was to implement this technique in the Lie group $SE\left(3\right)$ context. Given a set of noisy points in $SE\left(3\right)$ representing measurements on the trajectory of a mobile, one wants to find the geodesic that best fits those points in a Riemannian least squares sense. Finally, applications to simulated data are proposed to illustrate this work. The limitations of such a method and future perspectives are discussed. Full article

This paper reports an application of the Lie algebra $se\left(3\right)$ of the Euclidean group $SE\left(3\right)$, which is isomorphic to the theory of screws in the velocity and acceleration analyses of serial manipulators. The symbolic computation of the infinitesimal kinematics allows one to obtain algebraic expressions related to the kinematic characteristics of the end effector of the serial manipulator, while in the case of complex manipulators, numerical computations are preferred owing to the emergence of long terms. The algorithm presented enables the symbolic computation of the velocity and acceleration characteristics of the end effector in serial manipulators in order to allow the compact and efficient modeling of velocity and acceleration analyses of both parallel and serial robotic manipulators. Unlike other algebras, these procedures can be extended without significant effort to higher-order analyses such as the jerk and jounce. Full article