Search Results (603)

Multi-object tracking (MOT) has attracted increasing attention and achieved remarkable progress. However, accurately tracking objects with homogeneous appearance, heterogeneous motion, and heavy occlusion remains a challenge because of two problems: (1) missing association due to recognizing an object as background and (2) false prediction caused by the predominant utilization of linear motion models and the insufficient discriminability of object appearance representations. To address these challenges, this paper proposes a lightweight, generic, and appearance-independent MOT method with an unscented Kalman filter (UKF) on a Lie group called LUKF-Track. The method utilizes detection boxes across the entire range of scores in data association and matches objects across frames by employing a motion model, where the propagation and prediction of object states are formulated using a UKF on the Lie group. LUKF-Track achieves state-of-the-art results on three public benchmarks, MOT17, MOT20, and DanceTrack, which are characterized by highly nonlinear object motion and severe occlusions. Full article

For individuals with mental illness who experience multidimensional marginalization, the risks of encountering discrimination and receiving inadequate care are compounded. Artificial intelligence (AI) systems have propelled the provision of mental healthcare through the creation of digital mental health applications (DMHAs). DMHAs can be trained to identify specific markers of distress and resilience by incorporating community knowledge in machine learning algorithms. However, DMHAs that use rule-based systems and large language models (LLMs) may generate algorithmic bias. At-risk populations face challenges in accessing culturally and linguistically competent care, often exacerbating existing inequities. Creating equitable solutions in digital mental health requires AI training models that adequately represent the complex realities of marginalized people. This narrative review analyzes the current literature on digital mental health through an intersectional framework. Using an intersectional framework considers the nuanced experiences of individuals whose identities lie at the intersection of multiple stigmatized social groups. By assessing the disproportionate mental health challenges faced by these individuals, we highlight several culturally responsive strategies to improve community outcomes. Culturally responsive strategies include digital mental health technologies that incorporate the lived experience of individuals with intersecting identities while reducing the incidence of bias, harm, and exclusion. Full article

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

The past decade has seen growing applications of the information flow-based causality analysis, particularly with the concise formula of its maximum likelihood estimator. At present, the algorithm for its estimation is based on differential dynamical systems, which, however, may raise an issue for coarsely sampled time series. Here, we show that, for linear systems, this is suitable at least qualitatively, but, for highly nonlinear systems, the bias increases significantly as the sampling frequency is reduced. This study provides a partial solution to this problem, showing how causality analysis can be made faithful with coarsely sampled series, provided that the statistics are sufficient. The key point here is that, instead of working with a Lie algebra, we turn to work with its corresponding Lie group. An explicit and concise formula is obtained, with only sample covariances involved. It is successfully applied to a system comprising a pair of coupled Rössler oscillators. Particularly remarkable is the success when the two oscillators are nearly synchronized. As more often than not observations may be scarce, this solution, albeit partial, is very timely. Full article

The integration of an inertial navigation system (INS) with the Global Navigation Satellite System (GNSS) is crucial for suppressing the error drift of the INS. However, traditional fusion methods based on the extended Kalman filter (EKF) suffer from geometric inconsistency, leading to biased estimates—a problem markedly exacerbated under large initial misalignment angles. The invariant extended Kalman filter (IEKF) embeds the state in the Lie group SE₂(3) to establish a more consistent framework, yet two limitations remain. First, its standard update fails to synergize complementary error information within the left-invariant formulation, capping estimation accuracy. Second, velocity and position states converge slowly under extreme misalignment. To address these issues, a multiplicative federated IEKF (MF-IEKF) was proposed. A geometrically consistent state propagation model on SE₂(3) is derived from multiplicative IMU pre-integration. Two parallel, mutually inverse left-invariant error sub-filters (ML₁-IEKF and ML₂-IEKF) cooperate to improve overall accuracy. For large-misalignment scenarios, a short-term multiplicative right-invariant sub-filter is introduced to suppress initial position and velocity errors. Extensive Monte Carlo simulations and KITTI dataset experiments show that MF-IEKF achieves higher navigation accuracy and robustness than ML₁-IEKF. Full article

Surface crack detection and temporal evolution analysis are fundamental tasks in remote sensing and photogrammetry, providing critical information for slope stability assessment, infrastructure safety inspection, and long-term geohazard monitoring. However, current unmanned aerial vehicle (UAV)-based crack detection pipelines typically treat spatial detection and temporal change analysis as separate processes, leading to weak geometric consistency across time and limiting the interpretability of crack evolution patterns. To overcome these limitations, we propose the Longitudinal Crack Fitting Network (LCFNet), a unified and physically interpretable framework that achieves, for the first time, integrated time-series crack detection and evolution analysis from UAV remote sensing imagery. At its core, the Longitudinal Crack Fitting Convolution (LCFConv) integrates Fourier-series decomposition with affine Lie group convolution, enabling anisotropic feature representation that preserves equivariance to translation, rotation, and scale. This design effectively captures the elongated and oscillatory morphology of surface cracks while suppressing background interference under complex aerial viewpoints. Beyond detection, a Lie-group-based Temporal Crack Change Detection (LTCCD) module is introduced to perform geometrically consistent matching between bi-temporal UAV images, guided by a partial differential equation (PDE) formulation that models the continuous propagation of surface fractures, providing a bridge between discrete perception and physical dynamics. Extensive experiments on the constructed UAV-Filiform Crack Dataset (10,588 remote sensing images) demonstrate that LCFNet surpasses advanced detection frameworks such as You only look once v12 (YOLOv12), RT-DETR, and RS-Mamba, achieving superior performance (mAP_50:95 = 75.3%, F1 = 85.5%, and CDR = 85.6%) while maintaining real-time inference speed (88.9 FPS). Field deployment on a UAV–IoT monitoring platform further confirms the robustness of LCFNet in multi-temporal remote sensing applications, accurately identifying newly formed and extended cracks under varying illumination and terrain conditions. This work establishes the first end-to-end paradigm that unifies spatial crack detection and temporal evolution modeling in UAV remote sensing, bridging discrete deep learning inference with continuous physical dynamics. The proposed LCFNet provides both algorithmic robustness and physical interpretability, offering a new foundation for intelligent remote sensing-based structural health assessment and high-precision photogrammetric monitoring. Full article

In this paper we study the existence of Killing vector fields for right-invariant metrics on five-dimensional Lie groups. We begin by providing some explanation of the classification lists of the low-dimensional Lie algebras. Then we review some of the known results about Killing vector fields on Lie groups. We take as our invariant metric the sum of the squares of the right-invariant Maurer–Cartan one-forms, starting from a coordinate representation. A number of such metrics are uncovered that have one or more extra Killing vector fields, besides the left-invariant vector fields that are automatically Killing for a right-invariant metric. In each case the corresponding Lie algebra of Killing vector fields is found and identified to the extent possible on a standard list. The computations are facilitated by use of the symbolic manipulation package MAPLE. Full article

We survey the recent developments on quantum invariants of 3-manifolds and links: $\widehat{Z}$ and $F_L$. They are q-series invariants originated from mathematical physics, inspired by the categorification of a numerical quantum invariant—the Witten–Reshetikhin–Turaev (WRT) invariant—of 3-manifolds. They exhibit rich features, for example, quantum modularity, infinite-dimensional Verma module structures, and knot–quiver correspondence. Furthermore, they have connections to the 3d-3d correspondence and other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras. Full article

In this paper, we make one step further in the recognition of black box groups of Lie type: given a black box group encrypting a special linear group of dimension 2 over a finite field of an unknown odd characteristic, we construct a black box field and a polynomial time isomorphism from the special linear group of dimension 2 over this new field to the black box, which can be made polynomial time-reversible for small characteristics at the expense of constructing a look-up table for the prime field. Our result opens a way to constructing structural proxies for black box groups of Lie type. 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

This work investigates the classification of three-dimensional complete contact metric manifolds that are non-Sasakian and satisfy the relation $Q\xi =\sigma \xi$, focusing on those that support an almost-generalized $\mathit{Ƶ}$-soliton. In the scenario where $\sigma$ is constant, we prove that if a generalized $\mathit{Ƶ}$-soliton $(M^n,g,\delta ,\eta ,V,\mu ,\Lambda )$ satisfies the condition $g(V,\xi )=0$, then $M^n$ must be either an Einstein manifold or locally isometric to the Lie group $E(1,1)$. Comparable classifications are obtained for $(\kappa ,\mu ,\vartheta )$-contact metric manifolds. Furthermore, we explore situations in which the potential vector field aligns with the Reeb vector field. We then provide the corresponding structural characterizations. Full article

We study the controllability of right-invariant bilinear systems on the complex and quaternionic special linear groups $\mathrm{Sl}(n,\mathbb{C})$ and $\mathrm{Sl}(n,\mathbb{H})$. The analysis relies on the Lie saturate$\mathrm{LS}(\Gamma )$, which characterizes controllability through convexity and closure properties of attainable sets, avoiding explicit Lie algebra computations. For $\mathrm{Sl}(n,\mathbb{C})$ with a strongly regular diagonal control matrix, we show that controllability is equivalent to the irreducibility of the drift matrix A, a property verified by the strong connectivity of its associated directed graph. For $\mathrm{Sl}(n,\mathbb{H})$, we derive controllability criteria based on quaternionic entries and the convexity of ${\mathbb{T}}^2$-orbits, which provide efficient sufficient conditions for general n and exact ones in the $2\times 2$ case. These results link algebraic and geometric viewpoints within a unified framework and connect to recent graph-theoretic controllability analyses for bilinear systems on Lie groups. The proposed approach yields constructive and scalable controllability tests for complex and quaternionic systems. Full article

Massive open online courses (MOOCs) represent an innovative online learning paradigm that has garnered considerable popularity in recent years, attracting a multitude of learners to MOOC platforms due to their accessible and adaptable instructional structure. However, the elevated dropout rate in current MOOCs limits their advancement. Current dropout prediction models predominantly employ fixed-size convolutional kernels for feature extraction, which insufficiently address temporal dependencies and consequently demonstrate specific limitations. We propose a Lie Group-based feature context-local fusion attention model for predicting dropout in MOOCs. This model initially extracts shallow features using Lie Group machine learning techniques and subsequently integrates multiple parallel dilated convolutional modules to acquire high-level semantic representations. We design an attention mechanism that integrates contextual and local features, effectively capturing the temporal dependencies in the study behaviors of learners. We performed multiple experiments on the XuetangX dataset to evaluate the model’s efficacy. The results show that our method attains a precision score of 0.910, exceeding the previous state-of-the-art approach by 3.3%. Full article

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group $Ni{l}^4$ endowed with six left-invariant Lorentzian metrics. Combined with prior results, we achieve a complete classification of such hypersurfaces on 4-dimensional nilpotent Lie groups. The core of our approach lies in the explicit derivation and solution of the Codazzi tensor equations, which directly leads to the construction of these hypersurfaces and provides their explicit parametrizations. Our main results establish the existence of Codazzi hypersurfaces on $Ni{l}^4$, demonstrate the non-existence of totally geodesic hypersurfaces, specify the algebraic condition for a Codazzi hypersurface to become parallel, and provide their explicit parametrizations. This observation highlights fundamental differences between Lorentzian and Riemannian settings within hypersurface theory. This work thus clarifies the distinct geometric properties inherent to the Lorentzian cases on nilpotent Lie groups. Full article

This article presents a 2D pose estimation method for an omnidirectional mobile robot with Mecanum wheels, using an extended Kalman filter (EKF) formulated on the Lie group $\mathrm{SO}\left(2\right)$. The purpose is estimate the robot’s position and orientation by fusing angular velocity measurements from the wheel encoders with data from an IMU. Employing Lie algebra, the EKF provides a consistent and compact representation of rotational motion, improving prediction and update steps. The filter was implemented in ROS 1 and validated in simulation using Gazebo, with a reference trajectory and real measurements used for evaluation. The system delivers higher pose estimation precision, validating the effectiveness in rotational maneuvers. Full article