Search Results (171)

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

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 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

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

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix $MM$, that results from the integrability conditions of Killing’s equations. For $n=4$, the matrix $MM$ is of size $20\times 6$. In the case where $MM$ has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of $MM$ can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list. Full article

It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S² (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article

This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane ${\mathbb{R}}^2$ and the affine group $Af{f}_{+}\left(2\right)$. We systematically summarize recent advances, emphasizing how the geometric and algebraic structures inherent in low-dimensional Lie groups influence the formation, shape, and properties of control sets—maximal regions where controllability is maintained. Control sets with non-empty interiors are of particular interest as they characterize regions where the system can be steered between states via bounded inputs. The review highlights key results concerning the existence, uniqueness, and boundedness of these sets, including criteria based on the Ad-rank condition and orbit analysis. We also underscore the central role of the symmetry properties of Lie groups, which facilitate the systematic classification and description of control sets, linking the abstract mathematical framework to concrete, physically motivated applications. To illustrate the practical relevance of the theory, we present examples from mechanics, motion planning, and neuroscience, demonstrating how control sets naturally emerge in diverse domains. Overall, this work aims to deepen the understanding of controllability regions in low-dimensional Lie group systems and to foster future research that bridges geometric control theory with applied problems. Full article

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. 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 work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of $\mathbf{Grp}\left(\mathcal{C}\right)$ follows unconditionally from the completeness of the base category $\mathcal{C}$, cocompleteness requires $\mathcal{C}$ to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit limit and colimit constructions are provided, with colimits realized via coequalizers of relations induced by group axioms over free group objects. Applications demonstrate cocompleteness in topological groups, ordered groups, and group sheaves, while Lie groups serve as counterexamples revealing necessary analytic constraints—particularly the impossibility of equipping free groups on non-discrete manifolds with smooth structures. Further results include the inheritance of regularity when the free group functor preserves finite products, the existence of internal hom-objects in locally Cartesian closed settings, monadicity for locally presentable $\mathcal{C}$, and homotopical extensions where model structures on $\mathbf{Grp}\left(\mathcal{M}\right)$ reflect those of $\mathcal{M}$. This framework unifies classical category theory with geometric obstruction theory, resolving fundamental questions on exactness transfer and enabling new constructions in homotopical algebra and internal representation theory. Full article

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that $di{m}_K\left(L\right)=\mid L/K\mid <\infty$, the transcendence degree $trd(L/K)<\infty$ and $trd(L/K)=\infty$ with $difftrd(L/K)<\infty$, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension $L=K\left(G\right)$; then, there is a Galois correspondence between the intermediate fields $K\subset K^{\prime }\subset L$ and the subgroups $e\subset G^{\prime }\subset G$, provided that $K^{\prime }$ is stable under a Lie algebra $\Delta$ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

This paper establishes a functorial algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface $\Sigma$ of genus $g\ge 2$, for a complex reductive Lie group G, and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ of representations of its fundamental group with relatively compact image. The dimension formula $dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$, where ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_k$ are conjugacy classes in a maximal compact subgroup $K\subset G$, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups. Full article

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is invertible, we derive the algebraic soliton equations for left-invariant metrics and classify all left-invariant generalized cross-curvature solitons (for the generalized equation ${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg$) on the standard 3D Lorentzian Lie algebra types (unimodular Types Ia, Ib, II, and III and non-unimodular Types IV.1, IV.2, and IV.3). For each Lie algebra type, we state the necessary and sufficient algebraic conditions on the structure constants, provide explicit formulas for the soliton vector fields X (when they exist), and compute the soliton parameter $\lambda$ in terms of the structure constants and the parameter $\rho$. Our results include several existence families, explicit nonexistence results (notably for Type Ib and Type IV.3), and consequences linking the existence of left-invariant solitons with local conformal flatness in certain cases. The classification yields new explicit homogeneous generalized cross-curvature solitons in the Lorentzian setting and clarifies how the parameter $\rho$ modifies the algebraic constraints. Examples and brief geometric remarks are provided. Full article