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High-resolution remote sensing (HRRS) scene classification is based on the semantic information in the image and its contextual spatial correlation, labelling different semantic categories for different HRRS. Convolutional neural networks (CNN) have been widely studied and applied in remote sensing scene classification in recent years. However, most of the existing CNN models focus on the global and high-level semantic features, ignoring the shallower feature information. In addition, complex background information and variable scales lead to a series of problems with large intra-class differences and high inter-class similarities, which also brings challenges to scene classification. To address the above problems and challenges, a scenario classification model based on the contextual spatial attention and channel attention mechanism of Lie Group manifold space learning is proposed in this study. In this model, we fully explore the multi-scale features of the scene (shallower-level and high-level) and propose a novel contextual spatial attention mechanism and channel attention mechanism. Extensive experimentation was carried out on the Union Remote Sensing Image Data Set (URSIS), which improved by 7.13% compared with the classical model. The experimental results showed that compared with other state-of-the-art remote sensing scene classification models, our proposed method has achieved significant improvement in classification accuracy and performance. [ABSTRACT FROM AUTHOR]

We study the Gödel universe as the Lie group with the induced left-invariant sub-Lorentz structure defined by some proper subspace of the Lie algebra that generates it. We find the expressions for timelike and isotropic extremals in elementary functions by methods of geometric optimal control. Also, we prove that these extremals are not closed but as a rule not complete, having proper open subintervals of the real line as maximal connected domains. [ABSTRACT FROM AUTHOR]

As commonly known, standard time integration of the kinematic equations of rigid bodies modeled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using the R 3 × S O (3) Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. It also turns out that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly. [ABSTRACT FROM AUTHOR]

Let X be a compact complex manifold in Fujiki’s class C , i.e., admitting a big (1 , 1) -class [ α ] . Consider Aut (X) the group of biholomorphic automorphisms and Aut [ α ] (X) the subgroup of automorphisms preserving the class [ α ] via pullback. We show that X admits an Aut [ α ] (X) -equivariant Kähler model: there is a bimeromorphic holomorphic map σ : X ~ → X from a Kähler manifold X ~ such that Aut [ α ] (X) lifts holomorphically via σ . There are several applications. We show that Aut [ α ] (X) is a Lie group with only finitely many components. This generalizes an early result of Fujiki and Lieberman on the Kähler case. We also show that every torsion subgroup of Aut (X) is almost abelian, and Aut (X) is finite if it is a torsion group. [ABSTRACT FROM AUTHOR]

This introductory study suggests a formal basis for the interpretation of a continuous path in a connected matrix Lie group to be represented by the set of von Neumann ordinals which is a set-theoretical interpretation of natural numbers. In this study, it is aimed to relate the discrete recurrent structure of von Neumann ordinals to the exponential function. Since the Exponential function is fundamentally integrated into science and engineering literature this work aims to discover ties between the Exponential function and sets where, the Exponential function utilized in machine learning, loss functions; cryptography, key exchange and encryption algorithms; robotics, kinematics, trajectory planning; numerical analysis, discrete integration. Thus, the set theoretical interpretation of the exponential function has an interdisciplinary critical role. Throughout the article, necessary conjectures are postulated to interpret the rotations that form a smooth curve in terms of sets, namely von Neumann ordinals. Introduced formalizations covering Set existence axiom, unit element for set groups, interpretation of a smooth curve in terms of multiplication of exponentials, introduced a derivative operator to observe limited differentiable properties of the exponential function. [ABSTRACT FROM AUTHOR]

Studying the Gödel universe as a Lie group with left-invariant Lorentz metric and using the methods of geometric theory of the optimal control for the search of geodesics on Lie groups with left-invariant (sub-)Lorentz metrics, we find the expressions for timelike and isotropic geodesics in elementary functions. Also, we pay special attention to proving that the Gödel universe has no closed timelike or isotropic geodesics. [ABSTRACT FROM AUTHOR]

In this article, we prove that there are at least two new non-naturally reductive Ad(SO(l) × SO(k) × SO(k) × SO(k))-invariant Einstein metrics on compact simple Lie group SO(l + 3k) (k < l). It implies that every compact simple Lie group SO(n) (n > 12) admits at least 2 ([ n − 1 4 ] − 2) non-naturally reductive left invariant Einstein metrics. Moreover, we obtain that there are at least two families of invariant Einstein–Randers metrics on the compact Lie group SO(n) (n > 12). Besides, we construct non-naturally reductive left invariant (α, β) metrics on the Lie group SO(n) (n > 12). Finally, we examine the isometric problem for those Riemannian Einstein metrics. [ABSTRACT FROM AUTHOR]

This paper presents a new framework for corotational shell elements. The traditional corotational formulation uses local element frames, which greatly facilitates the calculation of elastic forces. However, all force vectors eventually need to be transformed into the global frame, resulting in the loss of invariance. The special Euclidean group SE(3) is introduced to describe the kinematics of the corotational shell element in the local frame. The equations of motion are established, in which the internal forces, inertial forces and tangent matrices are systematically derived in the SE(3) framework. The force vectors and their derivative matrices under the SE(3) description eliminate the effect of the rigid body motion, which is only related to the local deformation of the elements. Some examples are used to verify the validity and efficiency of the presented corotational shell element based on SE(3) to handle geometrically nonlinear problems. The results demonstrate that the SE(3) framework has higher computational efficiency with larger step size compared to the Lie group R3 × SO(3). According to the framework invariance brought by SE(3), a constant mass matrix during iterations is adopted to deal with the nonlinear problems with large rotation and small strain, which can significantly reduce the computational time. In summary, the results of the study show that the SE(3) framework has better characteristics and broader application prospects. [ABSTRACT FROM AUTHOR]

The chain group Ch(G) of a locally compact group G has one generator g ρ for each irreducible unitary G-representation ρ, a relation g ρ = g ρ ′ g ρ ″ whenever ρ is weakly contained in ρ ′ ⊗ ρ ″ , and g ρ * = g ρ − 1 for the representation ρ * contragredient to ρ. G satisfies chain-center duality if assigning to each g ρ the central character of ρ is an isomorphism of Ch(G) onto the dual Z (G) ̂ of the center of G. We prove that G satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. Müger's result compact groups satisfy chain-center duality. [ABSTRACT FROM AUTHOR]

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. [ABSTRACT FROM AUTHOR]

In this paper, we give the flag curvature formula of general (α, β)-metrics of Berwald type. We study conformally related (α, β)-metrics, especially general (α, β)-metrics that are conformally related to invariant (α, β)-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an (α, β)-metric is given and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related (α, β)-metrics. [ABSTRACT FROM AUTHOR]

Considering a representation space for a group of unimodular diag (1 ,   − 1 ,   − 1) -matrices, we construct several bases whose elements are eigenfunctions of Casimir infinitesimal operators related to a reduction in the group to some one-parameter subgroups. Finding the kernels of base transformation integral operators in terms of special functions, we consider the compositions of some of these transformations. Since composition is a 'closed' operation on the set of base transformations, we obtain some integral relations for the special functions involved in the above kernels. [ABSTRACT FROM AUTHOR]

The efficient fusion of global and local multi-scale features is quite important for remote sensing scene classification (RSSC). The scenes in high-resolution remote sensing images (HRRSI) contain many complex backgrounds, intra-class diversity, and inter-class similarities. Many studies have shown that global features and local features are helpful for RSSC. The receptive field of a traditional convolution kernel is small and fixed, and it is difficult to capture global features in the scene. The self-attention mechanism proposed in transformer effectively alleviates the above shortcomings. However, such models lack local inductive bias, and the calculation is complicated due to the large number of parameters. To address these problems, in this study, we propose a classification model of global-local features and attention based on Lie Group space. The model is mainly composed of three independent branches, which can effectively extract multi-scale features of the scene and fuse the above features through a fusion module. Channel attention and spatial attention are designed in the fusion module, which can effectively enhance the crucial features in the crucial regions, to improve the accuracy of scene classification. The advantage of our model is that it extracts richer features, and the global-local features of the scene can be effectively extracted at different scales. Our proposed model has been verified on publicly available and challenging datasets, taking the AID as an example, the classification accuracy reached 97.31%, and the number of parameters is 12.216 M. Compared with other state-of-the-art models, it has certain advantages in terms of classification accuracy and number of parameters. [ABSTRACT FROM AUTHOR]

An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR]

In this work, using some tools of Lie theory, we compute the Lie algebra of the Shearlet group regarding as a 3-fold semidirect product Lie group. As we will see, it is a 3-fold semidirect sum of Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, the kinematic models of the Strapdown Inertial Navigation System (SINS) and its errors on the SE (3) group in the Earth-Centered Inertial frame (ECI) are established. On the one hand, with the ECI frame being regarded as the reference, based on the joint representation of attitude and velocity on the SE (3) group, the dynamic of the local geographic coordinate system (n-frame) and the body coordinate system (b-frame) evolve on the differentiable manifold, respectively, and the high-order expansion of the Baker–Campbell–Haussdorff equation compensates for the non-commutative motion errors stimulated by strong maneuverability. On the other hand, the kinematics of the left- and right-invariant errors of the n-frame and the b-frame on the SE (3) group are separately derived, where the errors of the b-frame completely depend on inertial sensor errors, while the errors of the n-frame rely on position errors and velocity errors. In this way, the errors brought by the inconsistency of the reference coordinate system are tackled, and a novel attitude error definition is introduced to separate and decouple the factors affecting the dynamic of the n-frame errors and the b-frame errors for better attitude estimation. Through a turntable experiment and a car-mounted field experiment, the effectiveness of the proposed kinematic models in estimating attitude has been verified, with a remarkable improvement in yaw angle accuracy in the case of large initial misalignment angles, and the models developed have better robustness compared to the traditional SE (3) group-based model. [ABSTRACT FROM AUTHOR]

In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by A6,20-A6,27. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified. [ABSTRACT FROM AUTHOR]

We present the application of Lie group theory analysis with ζ -Riemann–Liouville fractional derivative (ζ -RLFD, for short) detailing the construction of infinitesimal prolongation to obtain Lie symmetries. In addition, it addresses the invariance condition without necessarily imposing that the lower limit of the fractional integral is fixed. We find an expression that expands the knowledge regarding the study of exact solutions for fractional differential equations. We apply the Leibniz-type rule for the derivative operator in question to build the prolongation. At last, we calculate the Lie symmetries of the generalized Burgers equation and fractional porous medium equation. [ABSTRACT FROM AUTHOR]

For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood centered-ball maximal-function of $f$ is given by the `supremum-norm':$$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a Lie group, $d$ is a left-invariant metric, and $\mu$ is a left Haar-measure (resp. right Haar-measure) are studied.

In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.

This paper proposes an improved initial alignment method for a strap-down inertial navigation system/global navigation satellite system (SINS/GNSS) integrated navigation system with large misalignment angles. Its methodology is based on the three-dimensional special Euclidean group and extended Kalman filter (SE2(3)/EKF) and aims to overcome the challenges of achieving fast alignment under large misalignment angles using traditional methods. To accurately characterize the state errors of attitude, velocity, and position, these elements are constructed as elements of a Lie group. The nonlinear error on the Lie group can then be well quantified. Additionally, a group vector mixed error model is developed, taking into account the zero bias errors of gyroscopes and accelerometers. Using this new error definition, a GNSS-assisted SINS dynamic initial alignment algorithm is derived, which is based on the invariance of velocity and position measurements. Simulation experiments demonstrate that the alignment method based on SE2(3)/EKF can achieve a higher accuracy in various scenarios with large misalignment angles, while the attitude error can be rapidly reduced to a lower level. [ABSTRACT FROM AUTHOR]

The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily. [ABSTRACT FROM AUTHOR]

In addressing the challenge of motion artifacts in Positron Emission Tomography (PET) lung scans, our study introduces the Triple Equivariant Motion Transformer (TEMT), an innovative, unsupervised, deep-learningbased framework for efficient respiratory motion correction in PET imaging. Unlike traditional techniques, which segment PET data into bins throughout a respiratory cycle and often face issues such as inefficiency and overemphasis on certain artifacts, TEMT employs Convolutional Neural Networks (CNNs) for effective feature extraction and motion decomposition.TEMT's unique approach involves transforming motion sequences into Lie group domains to highlight fundamental motion patterns, coupled with employing competitive weighting for precise target deformation field generation. Our empirical evaluations confirm TEMT's superior performance in handling diverse PET lung datasets compared to existing image registration networks. Experimental results demonstrate that TEMT achieved Dice indices of 91.40%, 85.41%, 79.78%, and 72.16% on simulated geometric phantom data, lung voxel phantom data, cardiopulmonary voxel phantom data, and clinical data, respectively. To facilitate further research and practical application, the TEMT framework, along with its implementation details and part of the simulation data, is made publicly accessible at https://github.com/yehaowei/temt. [ABSTRACT FROM AUTHOR]

In the present paper, we construct an example of almost cosymplectic 3-manifolds which is isomorphic to the group E(1, 1) of rigid motions of Minkowski 2-space. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the 3-dimensional Lie group (ℍ2 × ℝ, g) where g is a left-invariant Riemannian metric and we determine the Ricci solitons and Ricci bi-conformal vector fields on it. [ABSTRACT FROM AUTHOR]

The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given. [ABSTRACT FROM AUTHOR]

For a real-valued function f on a metric measure space (X, d, μ) the Hardy-Littlewood centered-ball maximal-function of f is given by the 'supremumnorm': ... In this note, we replace the supremum-norm on parameters r by Lp-norm with weight w on parameters r and define Hardy-Littlewood integral-function Ip,wf. It is shown that Ip,wf converges pointwise to Mf as p → ∞. Boundedness of the sublinear operator Ip,w and continuity of the function Ip,wf in case that X is a Lie group, d is a left-invariant metric, and μ is a left Haar-measure (resp. right Haar-measure) are studied. [ABSTRACT FROM AUTHOR]

We define the optimal constant Y (p 1 , p 2 ; G) of Young's convolution inequality as Y (p 1 , p 2 ; G) : = sup { ‖ ϕ 1 ∗ (ϕ 2 Δ 1 / p 1 ′ ) ‖ p ∣ ϕ 1 , ϕ 2 : G → C , ‖ ϕ 1 ‖ p 1 = ‖ ϕ 2 ‖ p 2 = 1 } for a locally compact group G and 1 ≤ p 1 , p 2 , p ≤ ∞ with 1 / p 1 + 1 / p 2 = 1 + 1 / p . Here, p ′ is the Hölder conjugate of p, ‖ · ‖ p is the L p -norm on a left Haar measure, and Δ : G → R > 0 is the modular function. The main result of this paper is that Y (p 1 , p 2 ; G) ≤ Y (p 1 , p 2 ; H) for any closed subgroup H ⊂ G . It follows from this inequality that Y (p 1 , p 2 ; G) ≤ Y (p 1 , p 2 ; R) dim G - r (G) for any connected Lie group G such that the center of the semisimple part is a finite group such as connected linear Lie groups and connected solvable Lie groups, where r(G) is the dimension of the maximal compact subgroups of G. [ABSTRACT FROM AUTHOR]

The paper deals with the twisted Sasaki metric on the unit tangent bundle of n-dimensional Riemannian manifold Mn. The main purpose of the research is to find deformations that preserve the existence harmonic left-invariant unit vector fields on 3-dimensional unimodular Lie groups G with the left invariant metric and harmonic maps G → T1G in case of twisted Sasaki metric on the unit tangent bundle. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map Mn → T1Mn are obtained. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map M² → T1M² with respect to some orthonormal frame are obtained. Left-invariant harmonic unit vector fields and harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric, using some orthonormal frame are described. Left-invariant harmonic unit vector fields which determine harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric in the particular case of twisted Sasaki metric, namely the vertical rescaled metric are classified. [ABSTRACT FROM AUTHOR]

We find geodesics, shortest arcs, cut loci, and first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups and . [ABSTRACT FROM AUTHOR]

This paper aims to classify a certain type of three-dimensional complete non-Sasakian contact manifold with specific properties, namely Q ξ = σ ξ and admitting Ricci–Bourguignon solitons. In the case of constant σ , the paper proves that if the potential vector field of the Ricci–Bourguignon soliton is orthogonal to the Reeb vector field, then the manifold is either Einstein or locally isometric to E(1, 1). Under a similar hypothesis, the paper shows that a (κ , μ , ϑ) -contact metric manifold is locally isometric to E(1, 1). Finally, the paper considers the scenario where the potential vector is pointwise collinear with the Reeb vector field and presents some results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie group. We mainly consider the solvable indecomposable six-dimensional Lie algebras with co-dimension two abelian nilradical that have an abelian complement. In dimension six, there are nineteen such algebras, namely, $ A_{6, 1} $–$ A_{6, 19} $ in Turkowski's list. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields, and finally we identify the symmetry Lie algebra from standard lists.

Casimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations. [ABSTRACT FROM AUTHOR]

In this article, a criterion for non-existence of closed (m, ρ)-quasi-Einstein metrics on (κ, μ)-almost coKähler manifolds is established. A similar result is stated for gradient (m, ρ)-quasi-Einstein metrics on three-dimensional (κ, μ)-almost coKähler manifolds. [ABSTRACT FROM AUTHOR]

Movements on surfaces of centered Euclidean spheres and changes between those with different radii mean complex multiplication in R 3 . Here, the Euclidean norm, which generates the spheres, is replaced with an inhomogeneous functional and a product is introduced in a geometric analogy. Because a change in the radius now leads to a change in the shape of the sphere, a three-dimensional dynamic complex structure is created. Statements about invariant probability densities, generalized uniform distributions on generalized spheres, geometric measure representations, and dynamic ball numbers associated with this structure are also presented. [ABSTRACT FROM AUTHOR]

We introduce the notions of (G, q)-opers and Miura (G, q)-opers, where G is a simply connected simple complex Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G, q)-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a qDE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects (q-differential equations). If g is simply laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra Uqĝ. However, if g is non-simply-laced, then these equations correspond to a different integrable model, associated to Uq Lĝ where Lĝ is the Langlands dual (twisted) affine algebra. A key element in this qDE/IM correspondence is the QQ-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category O of the relevant quantum affine algebra. [ABSTRACT FROM AUTHOR]

In this paper, we determine the bounded spherical functions associated with the Gelfand pair formed by the (4n + 3)-dimensional quaternionic Heisenberg group Hqn and the quaternionic unitary group Sp(n). We define the spherical Fourier transform and prove the Plancherel and inversion formulas. Additionally, we determine the heat kernel associated with the sub-Laplacian on Hqn. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie group. We mainly consider the solvable indecomposable six-dimensional Lie algebras with co-dimension two abelian nilradical that have an abelian complement. In dimension six, there are nineteen such algebras, namely, A6,1-A6,19 in Turkowski's list. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields, and finally we identify the symmetry Lie algebra from standard lists. [ABSTRACT FROM AUTHOR]

We study the analytic extension of a locally given Riemannian analytic metric to a metric of a nonextendable manifold. Various classes of locally isometric Riemannian analytic manifolds are studied. In each of these classes, the notion of the so-called pseudocomplete manifold is defined, which generalizes the notion of completeness of a manifold. A Riemannian analytic simply connected oriented manifold is said to be pseudocomplete if it is nonextendable and there exists no locally isometric orientation-preserving covering mapping with a simply connected Riemannian manifold. Among the pseudocomplete manifolds, we single out the "most symmetric" regular pseudocomplete manifolds. [ABSTRACT FROM AUTHOR]

We present the Julia package Manifolds.jl, providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase.jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds. [ABSTRACT FROM AUTHOR]

The primary purpose of this research is to work out a new action of Lie group through dual representation. In our paper we mention the basic definitions, we'll discuss the study of activity for Lie group upon Hom-space utilizing equivalence relationship between tensor product and Hom. Their measures will be studied on a structure consisting of four and five vector spaces. In the end we obtain new generalizations using action of dual representation for Lie group Ǥ. [ABSTRACT FROM AUTHOR]

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based Fe3O4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of Fe3O4, and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Let M3{M}^{3} be a strictly almost cosymplectic three-manifold whose Ricci operator is weakly ϕ\phi -invariant. In this article, it is proved that Ricci curvatures of M3{M}^{3} are invariant along the Reeb flow if and only if M3{M}^{3} is locally isometric to the Lie group E(1,1)E\left(1,1) of rigid motions of the Minkowski 2-space equipped with a left-invariant almost cosymplectic structure.

An important subclass among homogeneous spaces is formed by isotropically-faithful homogeneous spaces, in particular, this subclass contains all homogeneous spaces admitting invariant affine connection. An affine connection is equiaffine if it admits a parallel volume form. The purpose of the work is the local description of the three-dimensional homogeneous spaces that do not admit invariant equiaffine connections of nonzero curvature. We have concerned the case of the unsolvable Lie group of transformations. The basic notions, such as an isotropically-faithful pair, an invariant affine connection, curvature and torsion tensors, Ricci tensor, equiaffine connection are defined. A local study of homogeneous spaces is equivalent to the investigation of pairs consisting of a Lie algebra and its subalgebra. For three-dimensional homogeneous spaces of nonsolvable Lie groups that admit invariant connections of nonzero curvature only, it is determined under what conditions the space does not admit equiaffine connections. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character. A feature of the methods presented in the work is the application of a purely algebraic approach to the description of homogeneous spaces and connections on them. The results obtained in the work can be used in works on differential geometry, differential equations, topology, as well as in other areas of mathematics and physics, since many fundamental problems in these areas relate to the investigation of invariant objects on homogeneous spaces, the algorithms can be computerized and used for the solution of similar problems in large dimensions.

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension.

Considering a representation space for a group of unimodular diag(1, −1, −1)-matrices, we construct several bases whose elements are eigenfunctions of Casimir infinitesimal operators related to a reduction in the group to some one-parameter subgroups. Finding the kernels of base transformation integral operators in terms of special functions, we consider the compositions of some of these transformations. Since composition is a ‘closed’ operation on the set of base transformations, we obtain some integral relations for the special functions involved in the above kernels.

The efficient fusion of global and local multi-scale features is quite important for remote sensing scene classification (RSSC). The scenes in high-resolution remote sensing images (HRRSI) contain many complex backgrounds, intra-class diversity, and inter-class similarities. Many studies have shown that global features and local features are helpful for RSSC. The receptive field of a traditional convolution kernel is small and fixed, and it is difficult to capture global features in the scene. The self-attention mechanism proposed in transformer effectively alleviates the above shortcomings. However, such models lack local inductive bias, and the calculation is complicated due to the large number of parameters. To address these problems, in this study, we propose a classification model of global-local features and attention based on Lie Group space. The model is mainly composed of three independent branches, which can effectively extract multi-scale features of the scene and fuse the above features through a fusion module. Channel attention and spatial attention are designed in the fusion module, which can effectively enhance the crucial features in the crucial regions, to improve the accuracy of scene classification. The advantage of our model is that it extracts richer features, and the global-local features of the scene can be effectively extracted at different scales. Our proposed model has been verified on publicly available and challenging datasets, taking the AID as an example, the classification accuracy reached 97.31%, and the number of parameters is 12.216 M. Compared with other state-of-the-art models, it has certain advantages in terms of classification accuracy and number of parameters.

In this paper, the kinematic models of the Strapdown Inertial Navigation System (SINS) and its errors on the SE(3) group in the Earth-Centered Inertial frame (ECI) are established. On the one hand, with the ECI frame being regarded as the reference, based on the joint representation of attitude and velocity on the SE(3) group, the dynamic of the local geographic coordinate system (n-frame) and the body coordinate system (b-frame) evolve on the differentiable manifold, respectively, and the high-order expansion of the Baker–Campbell–Haussdorff equation compensates for the non-commutative motion errors stimulated by strong maneuverability. On the other hand, the kinematics of the left- and right-invariant errors of the n-frame and the b-frame on the SE(3) group are separately derived, where the errors of the b-frame completely depend on inertial sensor errors, while the errors of the n-frame rely on position errors and velocity errors. In this way, the errors brought by the inconsistency of the reference coordinate system are tackled, and a novel attitude error definition is introduced to separate and decouple the factors affecting the dynamic of the n-frame errors and the b-frame errors for better attitude estimation. Through a turntable experiment and a car-mounted field experiment, the effectiveness of the proposed kinematic models in estimating attitude has been verified, with a remarkable improvement in yaw angle accuracy in the case of large initial misalignment angles, and the models developed have better robustness compared to the traditional SE(3) group-based model.