Narrow Results

In this work, the dynamical structure for the extended equation is analyzed through unified Riccati equation expansion (UREE) and the Lie isomorphism method for the (3+1)-dimensional Wazwaz–Benjamin–Bona–Mahony (WBBM) equation. This equation represents the unidimensional propagation of short amplitude long waves on the water’s surface in a medium. These employed techniques are the most powerful and effective ways to obtain different sets of new and more generalized exact soliton solutions of the WBBM equation. Furthermore, what distinguishes this study from other studies is that it not only acquires a variety of analytical wave solutions for the studied models but also, demonstrates the interaction phenomena for these results as they propagate over time. Also, shows various meaningful graphs of the processes that provide valuable wisdom for understanding their behavior. The UREE method directly provides various new exact soliton solutions with some novel dynamical properties. We perform a detailed Lie symmetry analysis to governing equation that leaves the system invariant. The Lie group method explores six Lie isomorphism groups to study the WBBM equation. First, we find infinitesimal transformations employing the one-parameter Lie symmetry method. Second, we solve the infinitesimal generator and reduce the order of the equation. Moreover, we illustrate some two-dimensional (2D), three-dimensional (3D), and contour diagrams of the obtained results and compute the exact analytical solution utilizing the used methods. To find novel solutions, the Adomian method is also used, where the Adomian polynomials are utilized to deal with nonlinear terms. Variety of new analytical solutions with different types of dynamical behavior are analyzed by utilizing the computational software like Mathematica. These new analytical exact wave solutions are demonstrated in various dynamical structures of periodic wave soliton, interaction periodic wave and kink wave soliton, lump wave soliton, doubly soliton, multi-wave soliton, kink periodic, parabolic wave, multisoliton, traveling wave, and standing wave-shaped profiles.

Let $G$ be a finite permutation group acting on $\mathrm{\Omega }$. A base for $G$ is a subset $B\subseteq \mathrm{\Omega }$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $\mu (G)$, is the smallest cardinality of the support of a non-trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $\mu (G)b(G)\le nlogn$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant.

In this paper, using the Milnor-type theorem technique for each left-invariant symmetric $(0,2)$-tensor field $T$ on a five-dimensional nilpotent Lie group, we determine whether it is possible or not to find a left-invariant Riemannian metric $g$ and a constant $c$ so that $\mathrm{\text{Ric}}(g)=cT$, where $\mathrm{\text{Ric}}(g)$ is the Ricci curvature of $g$.

The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the connection-matrices further restricts the parameters spaces to differential manifolds such as the orthogonal group, the compact Stiefel manifold and its extensions. For these reasons, the instruments for characterizing and studying the behavior of learning equations for these particular networks are provided by the differential geometry of Lie groups. In particular, two sub-classes of the general Lie-group learning theories are studied in detail, dealing with first-order (gradient-based) and second-order (non-gradient-based) learning. Although the considered class of learning theories is very general, in the present paper special attention is paid to unsupervised learning paradigms.

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

Let $𝔾$ be a three-dimensional Lie group with a bi-invariant metric. Consider $\mathrm{\Omega }\subset 𝔾$ a strictly convex domain in $𝔾$. We prove that if $\mathrm{\Sigma }\subset \mathrm{\Omega }$ is a stable CMC free-boundary surface in $\mathrm{\Omega }$ then $\mathrm{\Sigma }$ has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z.287(1–2) (2017) 73–479] for the case where $𝔾={ℝ}^3$ and by R. Souam for the case where $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, excluding the possibility of $\mathrm{\Sigma }$ having three boundary components. Besides ${ℝ}^3$ and ${𝕊}^3$, our result also apply to the spaces ${𝕊}^1\times {𝕊}^1\times {𝕊}^1$, ${𝕊}^1\times {ℝ}^2$, ${𝕊}^1\times {𝕊}^1\times ℝ$ and $\mathrm{\text{SO}}(3)$. When $𝔾={𝕊}^3$ and $\mathrm{\Omega }$ is a geodesic ball with radius $r<\frac{\pi }{2}$, we obtain that if $\mathrm{\Sigma }$ is stable then $\mathrm{\Sigma }$ is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray $S$ on a Lie group $G$ satisfies that $G$ is commutative or $S$ is projective, then $S$ is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

In this research paper, we observed the Prandtl–Eyring magneto hydrodynamic fluid model (PE-MHDFM) by applying the Bayesian regularization scheme as backpropagated artificial neural networks (BRS-BANNs). Effect of suction/injection at the wall is the source of convective steady flow. The nonlinear partial differential equations (PDEs) of PE-MHDFM are converted into ordinary differential equations (ODE) by applying some suitable similarity transformation. These ODEs are solved by utilizing Lobatto IIIA numerical procedure to acquire the reference dataset for different scenarios of BRS-BANN. The reference dataset is used to design the solver BRS-BANN. Further, the performance of BRS-BANN is clarified by MSE results, error analysis plots, regression and error histogram. Moreover, the solution of PE-MHDFM is observed through the validation, training and testing procedures. It is observed that the best correlation between the targeted values outcomes of the study is matched effectively, which definitely authenticates the validity and reliability of the designed solver. Furthermore, the impacts on the velocity profile and temperature profile are examined by the variation of different physical quantities along with their comparison with state-of-the-art Lobatto IIIA numerical approach.

This work analyzes the two-dimensional flow of an incompressible magneto-hydrodynamic fluid over linear stretching sheet in the presence of suction or injection and convective boundary conditions. A scaling group transformation method is applied to the flow governing equations. The system remains invariant due to the relation between the transformation parameters. Upon finding three absolute invariants, third-order ordinary differential equations (ODEs) corresponding to momentum equation and second-order ODEs corresponding to energy and diffusion equations are derived. Shooting technique (R-K fourth-order) is applied to work out the flow equations numerically. MATLAB is used for the simulation and the results are exhibited through graphs. The computational results are validated with the published research work and a modest concurrence was found. The main outcome of this study is found to be that raising values of $f_{\mathrm{\text{si}}}$ and ${\mathrm{\Gamma }}_1$ decline the friction, whereas ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ show the opposite (increasing). The rising values of $f_{\mathrm{\text{si}}}$ and $M$ in addition to $M$ and ${\mathrm{\Gamma }}_2$ show a decline in friction factor. The Nusselt number values are improved as raising values of $Pr$ versus $f_{\mathrm{\text{si}}}$ and $Pr$ versus $M$. It is very clear the monotonically increasing $M$ versus $f_{\mathrm{\text{si}}}$ and strictly increasing $f_{\mathrm{\text{si}}}$ versus $M$ cases. It is very clear the mass-transfer rate is smoothly improved $L{e}_1$ versus $L{e}_2$ and strictly increased $L{e}_1$ versus ${\mathrm{\Gamma }}_1$.

In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

The iterative closet point (ICP) method is a dominant method for data registration that has attracted extensive attention. In this paper, a unified mathematical model of ICP based on Lie group representation is established. Under the framework, the registration problem is formulated into an optimization problem over a certain Lie group. In order to simplify the model and to reduce the dimension of parameter space, the translation part of geometric transformation is eliminated by calibrating the centers of two data sets under registration. As a result, a fast algorithm by solving an iterative linear system is designed for the optimization problem on Lie groups. Moreover, PCA and ICA methods are jointly applied to estimate the initial registration to achieve the global minimum. Finally, several illustrations and comparison experiments are presented to test the performance of the proposed algorithm.

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

This paper introduces a novel neural network approach based on Lie groups to effectively solve initial value problems of differential equations for nonlinear dynamical systems. Our method utilizes a priori knowledge inherent in the system, i.e. Lie group expressions, and employs a single-layer network structure with the essence of a multilayer perceptron (MLP). To validate the effectiveness of our approach, we conducted an extensive empirical study using various examples representing complex nonlinear dynamical systems. The research results demonstrate the outstanding performance and efficacy of our method, outperforming Neural Ordinary Differential Equations in terms of accuracy, convergence speed, and stability.

In this paper, we develop a general Lie group framework for analysis of kicking motion in humanoid soccer robots which aims to capture the multidimensional kicking information and hence to study how to develop more powerful and skilful kicking motion for humanoid soccer robots. To maintain dynamic stability while kicking is being performed, the zero-moment point (ZMP) is used to evaluate the performance of the humanoid kick. The proposed Lie-group-formulation-based compensation approach for force/torque sensing from the humanoid ankle has been implemented on, Robo-Erectus, our humanoid soccer robot. Humanoid kicking experiments have been conducted to verify the effectiveness of the proposed approach.

We show that, given a matrix Lie group and its Lie algebra , a 1-parameter subgroup of whose generator is the sum of an unperturbed matrix Â₀ and an analytic perturbation Â_♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of generated by a matrix belonging to the centralizer of Â₀ in . Both the similarity transformation and the matrix can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

A closed expression to the Baker–Campbell–Hausdorff (B-C-H) formula in SO(4) is given by making use of the magic matrix by Makhlin. As far as we know this is the first nontrivial example on (semi–) simple Lie groups summing up all terms in the B-C-H expansion.

On a Riemannian almost product manifold (M, P, g), we consider a linear connection preserving the almost product structure P and the Riemannian metric g and having a totally skew-symmetric torsion. We determine the class of the manifolds (M, P, g) admitting such a connection and prove that this connection is unique in terms of the covariant derivative of P with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the Kähler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold (G, P, g) constructed by a Lie group G.

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

We present here a range of considerations relating to the problem of Dirac for a quantifiable manifold (M, ω) to the dual a Lie algebra with a Poisson structure that is a Lie–Poisson structure minus.

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E⁴. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.