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A group theoretic analysis of the compressible Navier-Stokes equations of an ideal gas is carried out. The 12-dimensional Lie symmetry group is computed. The commutation table and the Levi decomposition of its Lie algebra are presented. The equations are reduced and self-similar one-, two- and three-dimensional solutions are computed. [ABSTRACT FROM AUTHOR]

Underwater vehicles heavily depend on the integration of inertial navigation with Doppler Velocity Log (DVL) for fusion-based localization. Given the constraints imposed by sensor costs, ensuring the optimization ability and robustness of fusion algorithms is of paramount importance. While filtering-based techniques such as Extended Kalman Filter (EKF) offer mature solutions to nonlinear problems, their reliance on linearization approximation may compromise final accuracy. Recently, Invariant EKF (IEKF) methods based on the concept of smooth manifolds have emerged to address this limitation. However, the optimization by matrix Lie groups must satisfy the "group affine" property to ensure state independence, which constrains the applicability of IEKF to high-precision positioning of underwater multi-sensor fusion. In this study, an alternative state-independent underwater fusion invariant filtering approach based on a two-frame group utilizing DVL, Inertial Measurement Unit (IMU), and Earth-Centered Earth-Fixed (ECEF) configuration is proposed. This methodology circumvents the necessity for group affine in the presence of biases. We account for inertial biases and DVL pole-arm effects, achieving convergence in an imperfect IEKF by either fixed observation or body observation information. Through simulations and real datasets that are time-synchronized, we demonstrate the effectiveness and robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We review recent progress regarding the index theory of operators defined on non-compact manifolds that can be modeled by Lie groupoids. The structure of a particular type of almost regular foliation is recalled and the construction of the corresponding accompanying holonomy Lie groupoid. Using deformation groupoids, K-theoretical invariants can be defined and compared. We summarize how questions in index theory are addressed via the geometrization made possible by the use of deformation groupoids. The discussion is motivated by examples and applications to degenerate PDE's, diffusion processes, evolution equations and geometry. [ABSTRACT FROM AUTHOR]

Cylindrically symmetric quantum mechanical systems with position dependent masses admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are thirty superintegrable and twelve maximally superintegrable ones. The arbitrary elements of the corresponding Hamiltonians (i.e.,masses and potentials) are presented explicitly. [ABSTRACT FROM AUTHOR]

AbstractThe Grigoriev-Meleshko Method, an indirect Lie group theory method, is used to derive the symmetry determining equations (SDEs) of the neutron transport equation (NTE) and the coupled delayed neutron precursor equations (DNPEs). A solution to the SDEs is a Lie group of transformations that can be used to reduce the order of the NTE and DNPEs or outright solve the equations. We found several solutions of the SDEs and worked through the mathematical algorithm to demonstrate relationships of instantiations of the NTE and its known solutions with the Lie groups. Examples of solutions include the Lie group that allows for the transformation of the differential form of the NTE to the integral form of the NTE; the Lie groups that permit Case’s solution; and the Lie group used to transform from the NTE to the $$\alpha $$α-eigenvalue form of the NTE. [ABSTRACT FROM AUTHOR]

We prove several rigidity results about the centralizer of a smooth diffeomorphism, concentrating on two families of examples: diffeomorphisms with transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. We classify all smooth diffeomorphisms with transitive centralizer: they are exactly the maps that preserve a principal fiber bundle structure, acting minimally on the fibers and trivially on the base. We also show that for any smooth, accessible isometric extension |$f_{0}\colon M\to M$| of an Anosov diffeomorphism of a nilmanifold, subject to a spectral bunching condition, any |$f\in \textrm{Diff}^{\infty }(M)$| sufficiently |$C^{1}$| -close to |$f_{0}$| has centralizer a Lie group. If the dimension of this Lie group equals the dimension of the fiber, then |$f$| is a principal fiber bundle morphism covering an Anosov diffeomorphism. Using the results of this paper, we classify the centralizer of any partially hyperbolic diffeomorphism of a |$3$| -dimensional, nontoral nilmanifold: either the centralizer is virtually trivial, or the diffeomorphism is an isometric extension of an Anosov diffeomorphism, and the centralizer is virtually |${{\mathbb{Z}}}\times{{\mathbb{T}}}$|⁠. [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]

This paper proposes a dead-reckoning (DR) method for vehicles using Lie theory. This approach treats the pose (position and attitude) and velocity of the vehicle as elements of the Lie group SE 2 (3) and follows the computations based on Lie theory. Previously employed DR methods, which have been widely used, suffer from cumulative errors over time due to inaccuracies in the calculated changes from velocity during the motion of the vehicle or small errors in modeling assumptions. Consequently, this results in significant discrepancies between the estimated and actual positions over time. However, by treating the pose and velocity of the vehicle as elements of the Lie group, the proposed method allows for accurate solutions without the errors introduced by linearization. The incremental updates for pose and velocity in the DR computation are represented in the Lie algebra. Experimental results confirm that the proposed method improves the accuracy of DR. In particular, as the motion prediction time interval of the vehicle increases, the proposed method demonstrates a more pronounced improvement in positional accuracy. [ABSTRACT FROM AUTHOR]

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula. [ABSTRACT FROM AUTHOR]

In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted K-theory of connected, simply-connected and simple compact Lie groups G, in both equivariant and nonequivariant settings. [ABSTRACT FROM AUTHOR]

Tactile servoing is an important technique because it enables robots to manipulate objects with precision and accuracy while adapting to changes in their environments in real-time. One approach for tactile servo control with high-resolution soft tactile sensors is to estimate the contact pose relative to an object surface using a convolutional neural network (CNN) for use as a feedback signal. In this paper, we investigate how the surface pose estimation model can be extended to include shear, and utilise these combined pose-and-shear models to develop a tactile robotic system that can be programmed for diverse non-prehensile manipulation tasks, such as object tracking, surface-following, single-arm object pushing and dual-arm object pushing. In doing this, two technical challenges had to be overcome. Firstly, the use of tactile data that includes shear-induced slippage can lead to error-prone estimates unsuitable for accurate control, and so we modified the CNN into a Gaussian-density neural network and used a discriminative Bayesian filter to improve the predictions with a state dynamics model that utilises the robot kinematics. Secondly, to achieve smooth robot motion in 3D space while interacting with objects, we used SE (3) velocity-based servo control, which required re-deriving the Bayesian filter update equations using Lie group theory, as many standard assumptions do not hold for state variables defined on non-Euclidean manifolds. In future, we believe that pose-and-shear-based tactile servoing will enable many object manipulation tasks and the fully-dexterous utilisation of multi-fingered tactile robot hands. [ABSTRACT FROM AUTHOR]

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

In this study, we consider model equations K(m,n) with fractional Caputo time derivatives. By applying the Lie group symmetry method, we determine all symmetries for these equations and present the reduced symmetric equations for the equation K(m,n) with fractional Caputo time derivatives. Furthermore, we obtain the exact solution for K(1,1) with the fractional Caputo time derivative and provide graphs depicting the behavior at different orders of the fractional time derivative. Additionally, by considering the symmetries of the equation, we establish the conservation laws for K(m,m) with the fractional Caputo time derivative. [ABSTRACT FROM AUTHOR]

The (2+1)-dimensional time-fractional chiral non-linear Schrödinger equation in physical media is considered in this paper. At the outset, the Lie group analysis is applied to build a set of infinitesimal generators for this equation with the aid of the Riemann–Liouville fractional derivatives. Consequently, the reduction for the considered equation into an ordinary differential equation of fractional order is obtained by using these generators and the ErdLélyi–Kober fractional operator. As a result of this reduction, we use power series analysis to get an analytical solution provided by a convergence analysis of the obtained solution. Furthermore, we construct a numerical solution based on hyperbolic functions using the fractional reduced differential transform method in the sense of Caputo fractional derivatives. Also, we detect absolute errors by performing a comparison between the exact and numerical solutions of the equation under study, while investigating the effect of fractional order α on the numerical solution. Finally, conservation laws are derived using the the formal Lagrangian and new conservation theorem. [ABSTRACT FROM AUTHOR]

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure. [ABSTRACT FROM AUTHOR]

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification G C of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. [ABSTRACT FROM AUTHOR]

In the paper, we describe a method for deriving generalized symmetries for a generic discrete quadrilateral equation that allows a Lax pair. Its symmetry can be interpreted as a flow along the tangent direction of its solution evolving with a Lie group parameter t. Starting from the spectral problem of the quadrilateral equation and assuming the eigenfunction evolves with the parameter t, one can obtain a differential-difference equation hierarchy, of which the flows are proved to be commuting symmetries of the quadrilateral equation. We prove this result by using the zero-curvature representations of these flows. As an example, we apply this method to derive symmetries for the lattice potential Korteweg–de Vries equation. [ABSTRACT FROM AUTHOR]

In this paper, we analyze submersions with Poisson fibres. These are submersions whose total space carries a Poisson structure, on which the ambient Poisson structure pulls back, as a Dirac structure, to Poisson structures on each individual fibre. Our "Poisson--Dirac viewpoint" is prompted by natural examples of Poisson submersions with Poisson fibres -- in toric geometry and in Poisson--Lie groups -- whose analysis was not possible using the existing tools in the Poisson literature. The first part of the paper studies the Poisson--Dirac perspective of inducing Poisson structures on submanifolds. This is a rich landscape, in which subtle behaviours abound, as illustrated by a surprising "jumping phenomenon" concerning the complex relation between the induced and the ambient symplectic foliations, which we discovered here. These pathologies, however, are absent from the well-behaved and abundant class of coregular submanifolds, with which we are mostly concerned here. The second part of the paper studies Poisson submersions with Poisson fibres -- the natural Poisson generalization of flat symplectic bundles. These Poisson submersions have coregular Poisson--Dirac fibres, and behave functorially with respect to such submanifolds. We discuss the subtle collective behavior of the Poisson fibres of such Poisson fibrations, and explain their relation to pencils of Poisson structures. The third and final part applies the theory developed to Poisson submersions with Poisson fibres which arise in Lie theory. We also show that such submersions are a convenient setting for the associated bundle construction, and we illustrate this by producing new Poisson structures with a finite number of symplectic leaves. Some of the points in the paper being fairly new, we illustrate the many fine issues that appear with an abundance of (counter-)examples. [ABSTRACT FROM AUTHOR]

In this paper, the new solitary wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation are obtained by Lie group symmetry method and the extended homoclinic test approach. Firstly, the equation can be reduced to (1+1)-dimensional partial differential equation by Lie group symmetry, and corresponding bilinear forms of the equation are given by symmetry functions. Secondly, the extended homoclinic test approach is employed to obtain the new kink type and singular solitary wave solutions. In addition, some new traveling and non-traveling wave solutions with arbitrary functions and oscillating tail are investigated through the special transformations for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. • The two different bilinear forms are obtained. • Some new solitary wave solutions are given. • Some new non-traveling wave solutions and their different behaviors are investigated. [ABSTRACT FROM AUTHOR]

We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher-rank simple Lie groups. Using Galois cohomology, we actually show that $\operatorname {SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups $E_{6(-14)}$ and $E_{7(-25)}$ constitute the complete list of higher-rank Lie groups admitting such examples. [ABSTRACT FROM AUTHOR]

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m -tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$ , the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$ , has p -torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p -torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ , which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group. [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]

We describe the inverse image of the Riemannian exponential map at a basepoint of a compact symmetric space as the disjoint union of so called focal orbits through a maximal torus. These are orbits of a subgroup of the isotropy group acting in the tangent space at the basepoint. We show how their dimensions (infinitesimal data) and connected components (topological data) are encoded in the diagram, multiplicities, Weyl group and lattice of the symmetric space. Obtaining this data is precisely what we mean by counting geodesics. This extends previous results on compact Lie groups. We apply our results to give short independent proofs of known results on the cut and conjugate loci of compact symmetric spaces. [ABSTRACT FROM AUTHOR]

Motivated by the issue of insufficient dynamic performance and tracking accuracy in SO(3)-based attitude tracking differentiators during large-angle maneuvers and complex trajectory tracking, a novel design approach for a three-degree-of-freedom attitude tracking differentiator within the SO(3) framework is proposed by incorporating second-order system theory and Lie group theory and improving the classical tracking differentiator. The kinematics model and error dynamics model of a rigid body on SO(3) are derived, and a reasonable virtual control input on SO(3) is constructed subsequently in order to achieve better dynamic response and tracking performance. Simulation and experimental results validate that the designed tracking differentiator could realize rapid and smooth convergence during large-angle maneuvers, and the initial large tracking error rapidly drops to near zero in a short period of time; additionally, it can also track expected time-varying curves well in complex trajectory tracking, with initial errors rapidly decreasing and maintaining at normal levels, demonstrating excellent tracking and control capabilities. There are strong application prospects for this new approach in addition to its theoretical significance. [ABSTRACT FROM AUTHOR]

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More precisely, we investigate some $L^2$ -estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot)\|_{L^2(G)}$ by further assuming the $L^1(G)$ -regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$ [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 give the exact contributions of Harish-Chandra transform, (1-lf)(>-.), of Schwartz functions f to the harmonic analysis of spherical convolutions and the corresponding V-Schwartz algebras on a connected semisimple Lie group G (with finite center). One of our major results gives the proof of how the TrombiVaradarajan Theorem enters into the spherical convolution transform of V-Schwartz functions and the generalization of this Theorem under the full spherical convolution map. [ABSTRACT FROM AUTHOR]

In this article, we find the subalgebra of the orthogonal Lie algebra corresponding to the DSER elementary orthogonal group on a quadratic space with a hyperbolic summand. [ABSTRACT FROM AUTHOR]

A semilinear quadratic equation of the form A i j (x) u i j = B i (x , u) u i + F (x , u) defines a metric A i j ; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities B i (x , u) , F (x , u). From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric A i j , while the second set serves as constraint equations, which select elements from the conformal algebra of A i j. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric A i j . In the present article, the nonlinear Poisson equation Δ g u − f (u) = 0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂ t . It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general 1 + (n − 1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δ g u − f (u) = 0 admits Lie point symmetries only when f (u) = k u , and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom. [ABSTRACT FROM AUTHOR]

We will prove that any non-empty open set in every complete connected metric space (X, d), where balls have compact closures, contains a paradoxical (uncountable) set relative to a non-supramenable connected Lie group that acts continuously and transitively on X. [ABSTRACT FROM AUTHOR]

If a Lie group admits a left invariant Randers metric of scalar flag curvature, then it is called of scalar Randers type. In this paper we determine all simply connected three dimensional Lie groups of scalar Randers type. It turns out that such groups must also admit a left invariant Riemannian metric with constant sectional curvature. [ABSTRACT FROM AUTHOR]

This is an I easier approach to implement into computer code i , because these equations [Eq. (95) of the paper] are simpler than the corresponding EOMs for { I i SB I n i sb , I i SB I n i sb }. CLARIFICATION ON THE NUMERICAL ALGORITHM USED TO PROPAGATE DYNAMICS We want to clarify the numerical algorithm we used to propagate the EOMs and generate all numerical results presented in the paper. In the above expressions, to compute HT ht , we use Eq. (C2) of the paper. [Extracted from the article]

This article, titled "Continuity Criteria for Locally Bounded Homomorphisms of Certain Lie Groups," discusses the continuity of locally bounded homomorphisms in Lie groups. The authors prove that a locally bounded homomorphism of a connected Lie group is continuous if and only if it is continuous on a closed supplementary subgroup. The article provides a theorem and a proof to support this claim. The research was partially supported by the Moscow Center for Fundamental and Applied Mathematics, and the author declares no conflict of interest. [Extracted from the article]

We present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based upon a new mapping formalism recently introduced by Runeson and Richardson [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the s u (N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich–Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom and derive an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connects the previous methods in a clear and concise manner. [ABSTRACT FROM AUTHOR]

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules. [ABSTRACT FROM AUTHOR]

Some generalizations of spin Sutherland models descend from 'master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson–Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the reduced systems possess the property of degenerate integrability on the dense open subset of the Poisson quotient space corresponding to the principal orbit type for the pertinent group action. After restriction to a smaller dense open subset, degenerate integrability on the generic symplectic leaves is demonstrated as well. The paper also contains a novel description of the reduced Poisson structure and a careful elaboration of the scaling limit whereby our reduced systems turn into the spin Sutherland models. [ABSTRACT FROM AUTHOR]

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p , q). In this paper, we study pseudo-Riemannian Lie groups (G , 〈 ⋅ , ⋅ 〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g , then dim ⁡ h ≤ max ⁡ { 0 , min ⁡ { p , q } − 2 }. Secondly, we classify trans-Lorentzian Lie groups (i.e., min ⁡ { p , q } = 2) with non-Killing left-invariant conformal vector fields, and prove that [ g , g ] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat. [ABSTRACT FROM AUTHOR]

We exhibit Anosov subgroups of |$\mathsf{SL}_{d}(\mathbb{R})$| that do not embed discretely in any rank- |$1$| simple Lie group of noncompact type, or indeed, in any finite product of such Lie groups. These subgroups are isomorphic to free products |$\Gamma * \Delta $|⁠ , where |$\Gamma $| is a uniform lattice in |$\textsf{F}_{4}^{(-20)}$| and |$\Delta $| is a uniform lattice in |$\textsf{Sp}(m,1)$|⁠ , |$m \geq 51$|⁠. [ABSTRACT FROM AUTHOR]

We study hypersurfaces of the four‐dimensional Thurston geometry Sol04$\mathrm{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of Sol04$\mathrm{Sol}^4_0$. We also give a closed expression for the Riemann curvature tensor of Sol04$\mathrm{Sol}^4_0$, using two integrable complex structures. [ABSTRACT FROM AUTHOR]

We introduce supergroup analogs of 3-manifold invariants Z ^ , also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the Z ^ invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup. [ABSTRACT FROM AUTHOR]

The present paper shows that local similarity theories, proposed for the strongly-stratified boundary layers, can be derived as invariant solutions defined under the Lie-group theory. A system truncated to the mean momentum and buoyancy equations is considered for this purpose. The study further suggests how similarity functions for the mean profiles are determined from the vertical fluxes, with a potential dependence on a measure of the anisotropy of the system. A time scale that is likely to characterize the transiency of a system is also identified as a non-dimensionalization factor. [ABSTRACT FROM AUTHOR]

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications. [ABSTRACT FROM AUTHOR]

Let G be a compact connected Lie group with Lie algebra \mathfrak {g}. We show that the category \operatorname {\mathbf {Loc}} _\infty (BG) of \infty-local systems on the classifying space of G can be described infinitesimally as the category {\operatorname {\mathbf {Inf}\mathbf {Loc}}} _{\infty }(\mathfrak {g}) of basic \mathfrak {g}-L_\infty spaces. Moreover, we show that, given a principal bundle \pi \colon P \to X with structure group G and any connection \theta on P, there is a differntial graded (DG) functor \begin{equation*} \mathscr {CW}_{\theta } \colon \mathbf {Inf}\mathbf {Loc}_{\infty }(\mathfrak {g}) \longrightarrow \mathbf {Loc}_{\infty }(X), \end{equation*} which corresponds to the pullback functor by the classifying map of P. The DG functors associated to different connections are related by an A_\infty-natural isomorphism. This construction provides a categorification of the Chern-Weil homomorphism, which is recovered by applying the functor \mathscr {CW}_{\theta } to the endomorphisms of the constant \infty-local system. [ABSTRACT FROM AUTHOR]