Starting from the Noether first theorem, we discuss a criterion for identifying physically consistent gravitational models. In particular, we demonstrate that applying the Lie derivative to a point-like Lagrangian makes it possible to determine the underlying symmetries, their associated generators, and the corresponding conserved quantities. This method is then generalized through its first prolongation and applied to several point-like Lagrangians, with special attention to f(R) gravity in a cosmological setting. In each example, the existence of symmetries results in a simplification of the dynamical system, enabling the integration of the equations of motion and the derivation of exact solutions. It is worth noticing that the existence of symmetries is always related to physically consistent models.

The main purpose of this article is to define two new concepts that will be known as a $n$-Lie derivation and $n$-Lie centralizer which is defined as follows: Let $R$ be a ring and an additive mapping $D:R\to R$ is called $n$-Lie derivation if $D([x^n,y^n])=[D(x^n),y^n]+[x^n,D(y^n)]$ for all $x,y\in R$ and an additive mapping $H:R\to R$ is called $n$-Lie centralizer if $H([x^n,y^n])=[H(x^n),y^n]=[x^n,H(y^n)]$ for all $x,y\in R$. In this article, we prove that under some torsion condition every $n$-Lie derivation is a Lie derivation and every $n$-Lie centralizer is Lie centralizer on semiprime ring.

We verify that the main result of Lin (2018) extends, mutatis mutandis, to $\ast$-Lie derivations on alternative $\ast$-algebras over the complex field. We state the corresponding theorem without reproducing the proof, since the argument is identical once the von Neumann analytic component is replaced by a Peirce-type nondegeneracy assumption natural to the alternative setting. As a byproduct, under primality (and 3-torsion freeness) this assumption becomes automatic in view of the result of Ferreira et al. (2018).

By employing the polar re-formulation, we show that there are no solutions of the Dirac equations in spherical symmetry when the spinor is required to satisfy the same symmetries as the spacetime via the Lie derivative.

Cohomology Groups and Lie Derivations of a Class of Lattice Algebras

Let G=G(A,B,M,N) be a generalized matrix algebra. A linear map Δ:G→G is called a Lie derivation at E∈G if Δ([U,V])=[Δ(U),V]+[U,Δ(V)] for all pairs U,V∈G such that UV=E. In this paper, we use techniques of matrix decomposition and algebraic identity analysis to fully characterize the general form of Lie derivations at E=e0000, where e0 is an arbitrary fixed element in A. Our main result establishes a necessary and sufficient condition for a Lie derivation at E=e0000 to be decomposable into the sum of a derivation of G and a center-valued linear map. This characterization significantly extends the classical results concerning global Lie derivations and provides a deeper insight into the local Lie-type behavior in operator algebras.

Purpose This paper investigates the topological and geometric properties of complete shrinking Riemann solitons (Mm, g, µ, V), extending classical results from Ricci solitons to the more general Riemann soliton setting. Design/methodology/approach We employ techniques from Riemannian geometry, including the analysis of the Riemann curvature tensor, Lie derivatives along vector fields and comparison theorems for geodesics. By establishing suitable inequalities on the divergence and norm of the soliton vector field V, we derive diameter bounds and compactness criteria. Findings We prove that any complete shrinking Riemann soliton is compact if the divergence of V satisfies divV ≤ −K1 and ||V || ≤ K2 for positive constants K1 and K2. Moreover, we show that the fundamental group of such a manifold is finite. Explicit diameter estimates in terms of K1, K2 and the soliton constant µ are provided. Originality/value These results generalize known compactness and finiteness theorems for Ricci solitons to the framework of Riemann solitons, offering new insights into their geometric structure and topological constraints.

Forward-flatness is a generalization of static feedback linearizability and a special case of a more general flatness concept for discrete-time systems. Recently, it has been shown that this practically quite relevant property can be checked by computing a unique sequence of involutive distributions which generalizes the well-known static feedback linearization test. In this paper, a dual test for forward-flatness based on a unique sequence of integrable codistributions is derived. Since the main mathematical operations for determining this sequence are the intersection of codistributions and the calculation of Lie derivatives of 1-forms, it is computationally quite efficient. Furthermore, the formulation with codistributions also facilitates a comparison with the existing discrete-time literature regarding the closely related topic of dynamic feedback linearization, which is mostly formulated in terms of 1-forms rather than vector fields. The presented results are illustrated by two examples.

Let [Formula: see text] be a unital ring in which [Formula: see text] is invertible, and let [Formula: see text] denote a generalized quaternion algebra over [Formula: see text]. In this paper, we describe the structure of Jordan derivations on [Formula: see text]. We then prove that every Jordan derivation is, in fact, a derivation on [Formula: see text]. Additionally, we characterize Lie derivations and Lie triple derivations of [Formula: see text] using Jordan derivations on [Formula: see text]. Also provide an example to illustrate our findings.

Identities pertaining to the de Rham codifferential [Formula: see text] in differential geometry are scattered in the literature. This paper gathers such formulas involving usual differential operators (Lie derivative, Schouten–Nijenhuis bracket, etc.), while adding some new ones using a natural extension of the interior product, to provide a compact handy summary.

Unmanned Aerial Vehicles (UAVs) are highly nonlinear and sophisticated systems that demand precise trajectory tracking in environments with uncertainties and disturbances. This research presents advanced nonlinear, adaptive, and artificial intelligence-based control strategies for UAVs. Beyond simulation, the strategies are experimentally evaluated on a coupled Two Degree of Freedom (2-DOF) Twin-rotor MIMO System (TRMS). The proposed strategies include Sliding Mode Control (SMC), Super Twisting (ST), BackStepping (BS), and Neuro-Adaptive SMC (NNSMC), all designed using a feedback linearized mathematical model of the system. System performance is enhanced by decoupling the TRMS into horizontal and vertical subsystems through Lie derivatives and diffeomorphism principles. A Uniform Robust Exact Differentiator (URED) estimates rotor speeds and recovers missing derivatives, while a nonlinear state feedback observer improves system observability and mitigates uncertainties and external wind gusts. Furthermore, ST and NNSMC-based laws reduce high-frequency oscillations in the control input of the first-order SMC law, resulting in improved transient response. The experimental results reveal that NNSMC significantly outperforms ST and BS in terms of trajectory tracking accuracy, transient performance, and integral performance indices for both pitch and yaw angles. These findings underscore the superior convergence performance and robustness of NNSMC, establishing it as a promising solution for precise TRMS control in real real-world environment.

The authors have designed a gyro-system for orientation (guidance) and stabilization, with two gimbals and a rotor in magnetic suspension (AMB—Active Magnetic Bearing) usable for self-guided rockets. The gyro-system (DGMSGG—double gimbal magnetic suspension gyro-system for guidance) orients and stabilizes the target coordinator’s axis (CT) and, at the same time, the AMB–rotor’s axis so that they overlap the guidance line (the target line). DGMSGG consists of two decoupled systems: one for canceling the AMB–rotor translations along the precession axes (induced by external disturbing forces), the other for canceling the AMB–rotor rotations relative to the CT-axis (induced by external disturbing moments) and, at the same time, for controlling the gimbals’ rotations, so that the AMB–rotor’s axis overlaps the guidance line. The nonlinear DGMSGG model is decomposed into two sub-models: one for the AMB–rotor’s translation, the other for the AMB–rotor’s and gimbals’ rotation. The second sub-model is described first by nonlinear state equations. This model is reduced to a second order nonlinear matrix—vector form with respect to the output vector. The output vector consists of the rotation angles of the AMB–rotor and the rotation angles of the gimbals. For this purpose, a differential geometry method, based on the use of the output vector’s gradient with respect to the nonlinear state functions, i.e., based on Lie derivatives, is used. This equation highlights the relative degree (equal to 2) with respect to the variables of the output vector and allows for the use of the dynamic inversion method in the design of stabilization and guidance controllers (of P.I.D.- and PD-types), as well as in the design of the related linear state observers. The controller of the subsystem intended for AMB–rotor’s translations control is chosen as P.I.D.-type, which leads to the cancellation of both its translations and its translation speeds. The theoretical results are validated through numerical simulations, using Simulink/Matlab models.

In this paper, we investigate the theory of Lie algebroid connections and Lie algebroid differential operators over a ringed space [Formula: see text] with [Formula: see text] being a sheaf of [Formula: see text]-algebras, where [Formula: see text] is a field of characteristic zero. We define Lie algebroid Lie derivative for an [Formula: see text]-module and hence proving the existence of Lie algebroid [Formula: see text]-derivations. We define the Lie algebroid jet bundles and construct the Lie algebroid connection algebra. We show that the category of Lie algebroid connections over a ringed space is equivalent to the category of modules over Lie algebroid connection algebra.

Abstract In this article, we consider some types of derivations in Banach algebras. In detail, we investigate the question of whether the superstability can be achieved under some conditions for some types of derivations, such as Jordan derivations, generalized Lie 2-derivations, and generalized Lie derivations.

Abstract This work delves into the characterization of mixed skew Lie and Jordan n-type ∗-derivations within the ∗-algebra 𝒜, which features a non trivial projection with unit I. In particular, if 𝒜 be an ∗-algebra, then every unital non linear mixed skew Lie and Jordan n-type derivations are additive ∗-derivations. Additionally, we explore the relevance of these derivations in the context of prime ∗-algebras, Von-Neuman algebras, Standard operator algebras.

Lie derivative algorithm for preserving geometry on cylindrical manifolds

Lie Derivations of Operator Algebras on Banach Spaces

In this work, we investigate the reach-avoid problem of a class of time-varying analytic systems with disturbances described by uncertain parameters. Firstly, by proposing the concepts of maximal and minimal reachable sets, we connect the avoidability and reachability with maximal and minimal reachable sets respectively. Then, for a given disturbance parameter, we introduce the evolution function for exactly describing the reachable set, and find a series representation of this evolution function with its Lie derivatives, which can also be regarded as a series function with respect to the uncertain parameter. Afterward, based on the partial sums of this series, over- and under-approximations of the evolution function are constructed, which can be realized by interval arithmetics with designated precision. Further, we propose sufficient conditions for avoidability and reachability and design a numerical quantifier elimination-based algorithm to verify these conditions; moreover, we improve the algorithm with a time-splitting technique. We implement the algorithms and use some benchmarks with comparisons to show that our methodology is both efficient and promising. Finally, we additionally extend our methodology to deal with systems with complex initial sets and time-dependent switchings. The performance of our extended method for these systems is also shown by four examples with comparisons and discussions.

Film-specific uniformity variations in packages are known to significantly diminish the effectiveness of the one-scan protocol, a commonly used film dosimetry method. This method universally adopts the reference dose-response with rescaling linearly from the relationship of the known dose and the unexposed state. This study aims to visualize and quantify the variation in unexposed film-specific uniformity in a package to evaluate the suitability of the reference dose response using machine-learning method. Fourteen EBT4 films (#00-#13) were selected from two lot packages. Nine grid-spaced 100 × 100 pixel (72 dpi) patches were obtained from the color images of EBT4 film sheet using a single scanner with landscape (scan A) and portrait (scan B) scan orientations. The reference patch was set at the center of film #00. For this study, multidimensional scaling (MDS) and Lie derivative image analysis (LDIA) were applied to the patch data with respect to the red (R)/green (G)/blue (B) channels. MDS is a suitable method for analyzing non-linear data with similarity, which provides a map of data objects according to a distance metric. LDIA directly detects the deviation vector field between image gradients. The film-specific uniformity was measured at 1/10000 scaled pixel value as a scalar distribution. The image flow field was obtained as a negative gradient of the scalar distribution. Two similarity metrics were defined for comparison with the reference patch: (1) MDSr (the distance parameter in the MDS map from the origin) and (2) Stot (summed S-value in each patch, where S-value represents the vorticity of the deviation vector field obtained via the Lie derivative). MDSr highly correlated with the absolute pixel value difference from the reference patch except for the blue channel in which a favorable package was detected for the reference dose response. Stot quantified the film-uniformity variation from the reference, independent of the dataset, and detected the unfavorable film state as Stot < 0.8 in the blue channel. We visualized and quantified the variation in film-specific uniformity in a lot package using MDS and LDIA, thereby quantitatively determining the unfavorable condition for applying the reference dose-response.

In this paper, we attempt to discuss the algebras of derivations, Lie derivations, generalized derivations and generalized Lie derivations of generalized quaternion ring [Formula: see text] where [Formula: see text] is a unital ring with a condition [Formula: see text]. We in principal give the results on the decomposition of above mentioned maps of [Formula: see text] in the terms of above mentioned maps of [Formula: see text] and inner derivation of [Formula: see text]. The proofs of the decomposition of Lie derivation of [Formula: see text] in terms of Lie derivation and Jordan derivation of [Formula: see text] and an inner derivation of [Formula: see text] and decomposition of generalized Lie derivation of [Formula: see text] in terms of a Lie derivation of [Formula: see text] and a left multiplier are given. Moreover, we also compute the matrix representation, admitted by the generalized Lie triple derivations of [Formula: see text] with respect to the standard basis.