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Exactly solvable Hamiltonians are useful in the study of quantum many-body systems using quantum computers. In the variational quantum eigensolver, a decomposition of the target Hamiltonian into exactly solvable fragments can be used for the evaluation of the energies via repeated quantum measurements. In this work, we apply more general classes of exactly solvable qubit Hamiltonians than previously considered to address the Hamiltonian measurement problem. The most general exactly solvable Hamiltonians we use are defined by the condition that within each simultaneous eigenspace of a set of Pauli symmetries, the Hamiltonian acts effectively as an element of a direct sum of so(N) Lie algebras and can, therefore, be measured using a combination of unitaries in the associated Lie group, Clifford unitaries, and mid-circuit measurements. The application of such Hamiltonians to decomposing molecular electronic Hamiltonians via graph partitioning techniques shows a reduction in the total number of measurements required to estimate the expectation value compared to previously used exactly solvable qubit Hamiltonians. [ABSTRACT FROM AUTHOR]

It is usually asserted that physical Hamiltonians for fermions must contain an even number of fermion operators. This is indeed true in electronic structure theory. However, when the Jordan–Wigner (JW) transformation is used to map physical spin Hamiltonians to Hamiltonians of spinless fermions, terms that contain an odd number of fermion operators may appear. The resulting fermionic Hamiltonian thus does not have number parity symmetry and requires wave functions that do not have this symmetry either. In this work, we discuss the extension of standard Hartree–Fock–Bogoliubov (HFB) theory to the number-parity-nonconserving case. These ideas had appeared in the literature before but, perhaps for lack of practical applications, had, to the best of our knowledge, never been employed. We here present a useful application for this more general HFB theory based on coherent states of the SO(2M + 1) Lie group, where M is the number of orbitals. We also show how using these unusual mean-field states can provide significant improvements when studying the JW transformation of chemically relevant spin Hamiltonians. [ABSTRACT FROM AUTHOR]

We define the concept of a flat pseudo-Riemannian F -Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent F -Lie algebras possessing one-dimensional light-cone subspaces. A similar result can be established for nilpotent Lie algebras equipped with flat scalar products of signature (2 , n − 2) where n ≥ 4. Furthermore, we use this technique to construct Poisson algebras exhibiting compatibility with flat scalar products. [ABSTRACT FROM AUTHOR]

This manuscript studies the global dynamics of a linear system on a disconnected Lie group. It shows that the connected components of the equilibria form a Lie subgroup, and the dynamics depends on the stability properties of the identity. The main result assures that if the identity is asymptotically stable the system decomposes as component systems with global attractor and component systems with dispersiveness. [ABSTRACT FROM AUTHOR]

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups. [ABSTRACT FROM AUTHOR]

In our previous work, we introduced a special type of Hermitian metrics called torsion-critical, which are non-Kähler critical points of the L^2-norm of Chern torsion over the space of all Hermitian metrics with unit volume on a compact complex manifold. In this short note, we restrict our attention to the class of compact Chern flat manifolds, which are compact quotients of complex Lie groups equipped with compatible left-invariant metrics. Our main result states that, if a Chern flat metric is torsion-critical, then the complex Lie group must be semi-simple, and conversely, any semi-simple complex Lie group admits a compatible left-invariant metric that is torsion-critical. [ABSTRACT FROM AUTHOR]

Let G be a real noncompact semisimple connected Lie group and let ρ : G ⟶ SL (V) be a faithful irreducible representation on a finite-dimensional vector space V over R . We suppose that there exists a scalar product g on V such that ρ (G) = K exp (p) , where K = SO (V , g) ∩ ρ (G) and p = Sym o (V , g) ∩ (d ρ) e (g) . Here, g denotes the Lie algebra of G, SO (V , g) denotes the connected component of the orthogonal group containing the identity element and Sym o (V , g) denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on P (V) arising from ρ . There is a corresponding G-gradient map μ p : P (V) ⟶ p . Using G-gradient map techniques, we prove that the unique compact G orbit O inside the unique compact U C orbit O ′ in P (V C) , where U is the semisimple connected compact Lie group with Lie algebra k ⊕ i p ⊆ sl (V C) , is the set of fixed points of an anti-holomorphic involutive isometry of O ′ and so a totally geodesic Lagrangian submanifold of O ′ . Moreover, O is contained in P (V) . The restriction of the function μ p β (x) : = ⟨ μ p (x) , β ⟩ , where ⟨ · , · ⟩ is an Ad (K) -invariant scalar product on p , to O achieves the maximum on the unique compact orbit of a suitable parabolic subgroup and this orbit is connected. We also describe the irreducible representations of parabolic subgroups of G in terms of the facial structure of the convex body given by the convex envelope of the image μ p (P (V)) . [ABSTRACT FROM AUTHOR]

We show that a compact open subgroup H of a simple algebraic p -adic group G is self-similar if and only if it is isotropic. [ABSTRACT FROM AUTHOR]

We introduce the notion of Θ -positivity in real semisimple Lie groups. This notion at the same time generalizes Lusztig's total positivity in split real Lie groups and invariant orders in Lie groups of Hermitian type. We show that there are four families of simple Lie groups which admit a positive structure relative to a subset Θ of simple roots, and investigate fundamental properties of Θ -positivity. We define and describe the positive and nonnegative unipotent semigroups and show that they give rise to a notion of positive n -tuples in flag varieties. [ABSTRACT FROM AUTHOR]

We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities. [ABSTRACT FROM AUTHOR]

In an influential 2005 paper, Michor and Mumford conjectured that in an infinite dimensional weak Riemannian manifold the vanishing of the geodesic distance is linked to the local unboundedness of the sectional curvature. We introduce infinite dimensional Hilbertian H-type groups equipped with any weak, graded, left invariant Riemannian metric. For these Lie groups we verify the above conjecture by showing that the vanishing of the geodesic distance and the local unboundedness of the sectional curvature coexist. We also observe that our class of weak Riemannian metrics yields the nonexistence of the Levi-Civita covariant derivative. [ABSTRACT FROM AUTHOR]

Let G be a Hermitian-type Lie group with the complexified Lie algebra 픤. We use L(λ) to denote a highest weight Harish-Chandra G-module with infinitesimal character λ. Let w be an element in the Weyl group W. We use Lw to denote a highest weight module with highest weight − wρ − ρ. In this paper we prove that there is only one Kazhdan–Lusztig right cell such that the corresponding highest weight Harish-Chandra modules Lw have the same associated variety. Then we give a characterization for those w such that Lw is a highest weight Harish-Chandra module and the associated variety of L(λ) will be characterized by the information of the Kazhdan–Lusztig right cell containing some special wλ. We also count the number of those highest weight Harish-Chandra modules Lw in a given Harish-Chandra cell. [ABSTRACT FROM AUTHOR]

In this study, we examine the Umbrella matrix group in Galilean space, which possesses the stochastic matrix property (i.e. leaves the 1 = 11...1T ∈ ℝ1n axis fixed) within matrix groups. In our examination, we first obtain the matrix Lie group and Lie algebra in G3 space and then generalize this result. Furthermore, we present the Cayley formula, which gives the transition between the SO(n) Lie group and Lie algebra, for the first time between the Galilean Umbrella matrix Lie group and Lie algebra. Then, we define a case of shear motion along the 1 axis with the help of a special Galilean transformation and generate rotated surfaces in Galilean space using certain curves. [ABSTRACT FROM AUTHOR]

The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) [11]. One of their results says that the Dirac cohomology of π completely determines π | K , where π is any irreducible unitarizable highest weight (g , K) module. This paper aims to develop this idea for the exceptional Lie groups E 6 (− 14) and E 7 (− 25) : we recover the K -spectrum of the Wallach modules from their Dirac cohomology. [ABSTRACT FROM AUTHOR]

We apply the quaternionic Jordan form to classify the nilpotent hypercomplex almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. Moreover, we determine which 12-dimensional simply connected hypercomplex almost abelian Lie groups admit lattices. Finally, for each integer n > 1 we construct infinitely many, up to diffeomorphism, (4 n + 4) -dimensional hypercomplex almost abelian solvmanifolds which are completely solvable. These solvmanifolds arise from a distinguished family of monic integer polynomials of degree n. [ABSTRACT FROM AUTHOR]

A compact symplectic manifold (M , ω) is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for (M , ω). This loosely means that there is a notion of harmonicity of differential forms in M , depending on ω alone, such that every de Rham cohomology class in has a ω -harmonic representative. In this article, we study two non-equivalent families of diagonal almost-abelian Lie algebras that admit a distinguished almost-Kähler structure and compute their cohomology explicitly. We show that they satisfy the hard-Lefschetz condition with respect to any left-invariant symplectic structure by exploiting an unforeseen connection with Kneser graphs. We also show that for some choice of parameters their associated simply connected, completely solvable Lie groups admit lattices, thereby constructing examples of almost-Kähler solvmanifolds satisfying the hard-Lefschetz condition, in such a way that their de Rham cohomology is fully known. [ABSTRACT FROM AUTHOR]

We consider the adjoint action of the symplectic Lie group Sp (2 n , C) on its Lie algebra sp (2 n , C). An element X ∈ sp (2 n , C) is called Ad Sp (2 n , C) -real if − X = Ad (g) X for some g ∈ Sp (2 n , C). Moreover, if − X = Ad (h) X for some involution h ∈ Sp (2 n , C) , then X ∈ sp (2 n , C) is called strongly Ad Sp (2 n , C) -real. In this paper, we prove that for every element X ∈ sp (2 n , C) , there exists a skew-involution g ∈ Sp (2 n , C) such that − X = Ad (g) X. Furthermore, we classify the strongly Ad Sp (2 n , C) -real elements in sp (2 n , C). We also classify skew-Hamiltonian matrices that are similar to their negatives via a symplectic involution. [ABSTRACT FROM AUTHOR]

The closed-form solutions of models provide operators and designers with a better understanding of how the systems perform practically, thus improving critical industrial production operations. Due to this importance, the case study at hand seeks to find closed-form solutions of the internal momentum and temperature variation during the filtration process to advance fluid purification. Lie group analysis is used to transform a system of equations representing the flow and heat transfer into a solvable system without changing the dynamics of the case study. The transformed solvable system is then integrated to find closed-form solutions of the internal velocity (momentum) and temperature variation. The obtained closed-form solutions are then used to analyze the effects of physical parameters arising from the process dynamics to find combinations of parameters that yield maximum permeates outflow. The analysis conveys that the internal fluid velocity increases when enhancing the permeation parameter and minimizing Reynolds number, wave speed parameter, and chamber height. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Janus symplectic group. We explore its various symmetries and its action on the elements of the dual of its Lie algebra, called torsors. Special attention is given to the charge symmetry, which highlights the matter–antimatter duality within both sets of components. [ABSTRACT FROM AUTHOR]

Given fusion categories of loop group representations at level l , we construct subfactors N ⊂ M of depth d which satisfy the following conditions : (1) d = l for type A n , C n , B 2 and β ⋅ l ≤ d ≤ l for the other types with β uniquely determined by the type. (2) The fusion ring of the associated bimodules is isomorphic to the one of loop group at level l. [ABSTRACT FROM AUTHOR]

In this paper, we are focusing on entanglement control problem in a three-qubit system. We demonstrate that vector representation of entanglement, associated with SO(6) representation of SU(4) two-qubit group, can be used to solve various control problems analytically including (i) the transformation between a W-type states and GHZ state, and (ii) manipulating bipartite concurrences and three-tangle under a restricted access to only two qubits, and (iii) designing USp(4)-type quaternionic operations and quantum states. [ABSTRACT FROM AUTHOR]

Let N be a simply connected nilpotent Lie group, and let K be a connected compact subgroup of the automorphism group, A u t (N) , of N. Let G : = K ⋉ N be the semidirect product (of K and N). Let ⊃ be the respective Lie algebras of G and K and q : ∗ → ∗ be the natural projection. It was pointed out by Lipsman, that the unitary dual G ̂ of G is in one-to-one correspondence with the space of admissible coadjoint orbits ‡ / G (see [R. L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl.59 (1980) 337–374]). Let π ∈ G ̂ be a generic representation of G and let τ ∈ K ̂. To these representations we associate, respectively, the admissible coadjoint orbit G ⊂ ∗ and K ⊂ ∗ (via the Lipsman's correspondence). We denote by χ ( G , K) the number of K -orbits in G ∩ q − 1 ( K) , which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman's orbit method suggests that the multiplicity m π (τ) of an irreducible K -module τ occurring in the restriction π | K could be read from the coadjoint action of K on G ∩ q − 1 ( K). Under some assumptions on the pair (K , N) , we prove that for a class of generic representations π ∈ G ̂ , one has m π (τ) ≠ 0 ⇒ χ ( G , K) ≠ 0. Moreover, we show that the Corwin–Greenleaf multiplicity function is bounded (≤ 1) for a special class of subgroups of G. Finally, we give a necessary and sufficient conditions to obtain a nonzero multiplicity ( m π (τ λ) ≠ 0). [ABSTRACT FROM AUTHOR]

We study the relation between two special classes of Riemannian Lie groups G with a left-invariant metric g : The Einstein Lie groups, defined by the condition Ric g = c g , and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups (G , g) are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the G × K -invariant geodesic orbit metrics on Lie groups G for a wide class of subgroups K that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds. [ABSTRACT FROM AUTHOR]

The initial attitude error is challenging to satisfy the requirements of the linear model due to the complex nature of the ocean environment. This presents a challenge in the transfer alignment of the ship. In order to enhance the precision and velocity of ship transfer alignment, as well as to streamline the alignment processes, this paper proposes a transfer alignment methodology based on the Earth-Centered Earth-Fixed (ECEF) frame special Euclidean group (SE(3)) matrix Lie group. After introducing the two navigation states, velocity and attitude, from the ECEF frame into SE(3), the nonlinear inertial navigation system error state model and its corresponding measurement equations are derived based on the mapping relationship between the Lie groups and Lie algebra. The method effectively solves the error problem due to linear approximation in the traditional transfer alignment method, and applies to misalignment angles of arbitrary scale. The simulation results verify the effectiveness and rapidity of the proposed alignment method in the case of arbitrary misalignment angles. [ABSTRACT FROM AUTHOR]

Carlen–Jauslin–Lieb–Loss Carlen et al. (Int Math Res Not 24:18604–18612, 2021) recently observed a remarkable property of a convolution inequality, namely the convolution inequality f ≥ f ∗ f for real-valued f ∈ L 1 (R n) implies that f is non-negative and has a non-trivial L 1 -bound. In this note, we improve their result by proving that these two consequences are still valid under the weaker assumption in terms of the N times iterated convolution f ≥ f ∗ f ∗ ⋯ ∗ f . Moreover, we extend these results to the convolution on some class of Lie groups especially including the Heisenberg group H n . We then apply our result on the iterated convolution on R n to the study of some integro–differential equations which generalise the equation investigated in Carlen–Jauslin–Lieb Carlen et al. (Pure Appl Anal 2:659–984, 2020) [ABSTRACT FROM AUTHOR]

In this paper, we present the Euler–Lagrange and Hamilton’s equations for a system whose configuration space is a unified product Lie group G = M⋈γH, for some γ : M × M → H. By reduction, then, we obtain the Euler–Lagrange-type and Hamilton’s-type equations of the same form for the quotient space M≅G/H, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler–Poincaré-type and Lie–Poisson-type equations on the corresponding quotient 픪≅픤/픥 of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the nth order iterated tangent group T(n)G of a Lie group G as an extension of the nth order tangent group TnG of the same type. More precisely, 픤 is the Lie algebra of G, T(n)G≅픤×2n−1−n⋈ γTnG for some γ : 픤×2n−1−n × 픤×2n−1−n → TnG. We thus obtain the nth order Euler–Lagrange (and then the nth order Euler–Poincaré) equations over TnG by reduction from those on T(Tn−1G). Finally, we illustrate our results in the realm of the Kepler problem, and the nonlinear tokamak plasma dynamics. [ABSTRACT FROM AUTHOR]

Given any compact homogeneous space H/K with H simple, we consider the new space M = H × H/ΔK, where ΔK denotes diagonal embedding, and study the existence, classification and stability of H × H-invariant Einstein metrics on M, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on M for most spaces H/K such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of 픨 is a multiple of the Killing form of 픥 (e.g., K simple), a class which contains 17 families and 50 individual examples. A complete classification is obtained in the case when H/K is an irreducible symmetric space and K is simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on M = H × H/ΔK (none of which is Einstein), obtaining that the standard metric is a global minimum. [ABSTRACT FROM AUTHOR]

AbstractLet G be a simple complex Lie group, Gekhtman, Shapiro, and Vainshtein made a conjecture: given a Belavin-Drinfeld triple for G, there exists a classification of regular cluster structures on G that is completely parallel to the Belavin-Drinfeld classification of solutions of the classical Yang-Baxter equation. This paper confirms the conjecture for non simply-laced complex Lie group SO(3,C) and its Drinfeld double D(SO(3,C)). We construct the corresponding cluster structure in the ring of regular functions on D(SO(3,C)). [ABSTRACT FROM AUTHOR]

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 Ric(g) = cT, where Ric(g) is the Ricci curvature of g. [ABSTRACT FROM AUTHOR]

To enhance the positioning accuracy of autonomous underwater vehicles (AUVs), a new adaptive filtering algorithm (RHAUKF) is proposed. The most widely used filtering algorithm is the traditional Unscented Kalman Filter or the Adaptive Robust UKF (ARUKF). Excessive noise interference may cause a decrease in filtering accuracy and is highly likely to result in divergence by means of the traditional Unscented Kalman Filter, resulting in an increase in uncertainty factors during submersible mission execution. An estimation model for system noise, the adaptive Unscented Kalman Filter (UKF) algorithm was derived in light of the maximum likelihood criterion and optimized by applying the rolling-horizon estimation method, using the Newton–Raphson algorithm for the maximum likelihood estimation of noise statistics, and it was verified by simulation experiments using the Lie group inertial navigation error model. The results indicate that, compared with the UKF algorithm and the ARUKF, the improved algorithm reduces attitude angle errors by 45%, speed errors by 44%, and three-dimensional position errors by 47%. It can better cope with complex underwater environments, effectively address the problems of low filtering accuracy and even divergence, and improve the stability of submersibles. [ABSTRACT FROM AUTHOR]

Let G be a finite permutation group acting on Ω. A base for G is a subset B ⊆ Ω 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 μ(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 μ(G)b(G) ≤ nlog n, where n is the degree of G. This bound is best possible, up to a multiplicative constant. [ABSTRACT FROM AUTHOR]

In this research, we explore a finite horizon optimal stabilization and tracking control scheme for the dynamical model of a 6‐DOF Autonomous Underwater Vehicle (AUV). Dynamical equations of the AUV are represented in a Lie group (SE(3)$$ SE(3) $$) framework, encompassing both translational and rotational motions. Utilizing a left Lie group action on SE(3)$$ SE(3) $$, we define error function for velocities via a right transport map to effectively address optimal trajectory tracking. The optimal control objective is formulated as a trade‐off problem, aiming to minimize both errors and control effort simultaneously. Left action on SE(3)$$ SE(3) $$ yields the left trivialized Hamiltonian function from which the concomitant state and costate dynamical equations are derived using Pontryagin's Minimum Principle (PMP). Consequently, the resulting two‐point boundary value problem is solved to obtain optimal trajectories. We demonstrate the optimality of the resulting solution obtained from the derived control law. For ensuring boundedness in the presence of small disturbances, this study incorporates the effects of internal parametric uncertainties associated with added mass and inertia components, along with the influence of external disturbances induced by ocean currents. Through simulation validations, we confirm the alignment of our results with the theoretical developments, demonstrating that the proposed control law effectively mitigates both parametric uncertainties and ocean current disturbances. [ABSTRACT FROM AUTHOR]

In this paper, I define and classify the algebraic Schouten solitons associated with the Bott connection on three-dimensional Lorentzian Lie groups with three different distributions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic conditions of algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Furthermore, we provide the complete classification for these algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the algebraic Schouten solitons. The main results indicate that G 4 does not possess algebraic Schouten solitons related to the perturbed Kobayashi–Nomizu connection, G 1 , G 2 , G 3 , G 6 , and G 7 possess algebraic Schouten solitons, and the result for G 5 is trivial. [ABSTRACT FROM AUTHOR]

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether's theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether's theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations. [ABSTRACT FROM AUTHOR]

We enhance the Thompson–Smith construction [34] , [35] , [32] , [33] of the Thompson sporadic simple group E. As a result, (a) we obtain a conceptual explanation of the dichotomy between Lie and finite cases in terms of representations of an (L 2 (8) : 3) -subgroup; (b) along with 2 5 ⋅ L 5 (2) and 2 + 1 + 8 ⋅ A 9 , we construct in E subgroups D 4 3 (2) : 3 , (F 21 × L 3 (2)) : 2 , (G 2 (3) × 3) : 2 and U 3 (8) : 6 ; (c) we consider the coset geometry of the above subgroups and identify the corresponding geometric presentation with that by Havas–Soicher–Wilson [14] ; (d) this amounts to a new geometric construction and uniqueness proof for E ; (e) we deliver a self-contained construction of the Dempwolff–Thompson orthogonal decomposition on the E 8 -Lie algebra, along with the stabiliser 2 5 + 10 ⋅ L 5 (2) of this decomposition in the Lie group E 8 (C). The project was accomplished within a general approach to finite simple groups through their simply connected geometries [19]. The work was further stimulating by the recent reappearance of E in the Moonshine context [2] , [13]. [ABSTRACT FROM AUTHOR]

This paper introduces a novel closed-form coordinate-free expression for the higher-order Cayley transform, a concept that has not been explored in depth before. The transform is defined by the Lie algebra of three-dimensional vectors into the Lie group of proper orthogonal Euclidean tensors. The approach uses only elementary algebraic calculations with Euclidean vectors and tensors. The analytical expressions are given by rational functions by the Euclidean norm of vector parameterization. The inverse of the higher-order Cayley map is a multi-valued function that recovers the higher-order Rodrigues vectors (the principal parameterization and their shadows). Using vector parameterizations of the Euler and higher-order Rodrigues vectors, we determine the instantaneous angular velocity (in space and body frame), kinematics equations, and tangent operator. The analytical expressions of the parameterized quantities are identical for both the principal vector and shadows parameterization, showcasing the novelty and potential of our research. [ABSTRACT FROM AUTHOR]

In this study, the Kundu–Eckhaus represents the propagation of femtosecond light pulse in optical fibers is solved using symmetry technique. As a result, a finite Lie group of four vector fields is yielded. Then a similarity transformation is obtained using a linear combination of the four vector fields, which considers the general transformation for any used transformation before. According to that, the Kundu–Eckhaus is reduced to an auxiliary equation. By solving the auxiliary equation many novel solitary wave solutions were obtained. Finally, the propagation of the amplitude bright soliton is discussed, and it was found that the chirp wave depends on the amplitude function and is affected by the sign of the Raman parameter. [ABSTRACT FROM AUTHOR]

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity map. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-Müller-Xie, on quasi-isometries of these solvable groups. [ABSTRACT FROM AUTHOR]

Let G be an acylindrically hyperbolic group on a ı-hyperbolic space X. Assume there exists M such that for any finite generating set S of G, the set SM contains a hyperbolic element on X. Suppose that G is equationally Noetherian. Then we show the set of the growth rates of G is well ordered. The conclusion was known for hyperbolic groups, and this is a generalization. Our result applies to all lattices in simple Lie groups of rank 1, and more generally, relatively hyperbolic groups under some assumption. It also applies to the fundamental group, of exponential growth, of a closed orientable 3-manifold except for the case that the manifold has Sol-geometry. [ABSTRACT FROM AUTHOR]

Kapovich, Leeb and Porti (2014) gave several new characterizations of Anosov representations ! G, including one where geodesics in the word hyperbolic group map to "Morse quasigeodesics" in the associated symmetric space G=K. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples. [ABSTRACT FROM AUTHOR]

This paper continues the study of two numbers that are associated with Lie groups. The first number is \(N(G,m)\), the number of conjugacy classes of elements in \(G\) whose order divides \(m\). The second number is \(N(G,m,s)\), the number of conjugacy classes of elements in \(G\) whose order divides \(m\) and which have \(s\) distinct eigenvalues, where we view \(G\) as a matrix group in its smallest-degree faithful representation. We describe systematic algorithms for computing both numbers for \(G\) a connected and simply-connected exceptional Lie group. We also provide explicit results for all of \(N(G,m)\), \(N(G_2,m,s)\), and \(N(F_4,m,s)\). The numbers \(N(G,m,s)\) were previously known only for the classical Lie groups; our results for \(N(G,m)\) agree with those already in the literature but are obtained differently.Mathematics Subject Classifications: 05A15, 05E16, 22E40, 22E10, 22E15Keywords: Lie groups, conjugacy classes, element of finite order, Burnside's lemma

In the present paper, we develop a general theory for the reduction of systems with Jacobi multiplier and volume-preserving symmetries on $ n $-manifolds ($ n\geq2 $). If the system has $ r $ commuting volume-preserving symmetries, we can reduce it to another system with a Jacobi multiplier, where the solutions of this reduced system are determined by the solutions of its first $ n-r $ equations. Specifically, if the system has $ n-2 $ such symmetries, then the solutions of the transformed equations are determined by the solutions of its first two equations, and these first two equations constitute a one-degree-of-freedom Hamiltonian system. We also use geometric methods to provide a local way to find the complete first integrals of the system. Additionally, we provide examples illustrating that using our method to find first integrals is equivalent to solving certain partial differential equations. Finally, we used a KAM-type theorem to investigate the stability of the solutions of this system. [ABSTRACT FROM AUTHOR]

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces. [ABSTRACT FROM AUTHOR]

Let $ X = G/\Gamma $, where $ G $ is a Lie group and $ \Gamma $ is a uniform lattice in $ G $, and let $ O $ be an open subset of $ X $. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape $ O $ on average with frequency $ \delta $, where $ 0 < \delta \le 1 $. [ABSTRACT FROM AUTHOR]

We consider a class of nonvariational degenerate elliptic operators of the kind L u = ∑ i , j = 1 m a i j (x) X i X j u where { a i j (x) } i , j = 1 m is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole R n (n > m), possibly discontinuous but satisfying a VMO assumption, and X 1 ,... , X m are real smooth vector fields satisfying Hörmander rank condition in the whole R n and 1-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global W X 2 , p a-priori estimates, for every p ∈ (1 , ∞) , of the kind: ‖ u ‖ W X 2 , p (R n) ≤ c { ‖ L u ‖ L p (R n) + ‖ u ‖ L p (R n) } for every u ∈ W X 2 , p (R n). We also prove higher order estimates and corresponding regularity results: if a i j ∈ W X k , ∞ (R n) , u ∈ W X 2 , p (R n) , L u ∈ W X k , p (R n) , then u ∈ W X k + 2 , p (R n) and ‖ u ‖ W X k + 2 , p (R n) ≤ c { ‖ L u ‖ W X k , p (R n) + ‖ u ‖ L p (R n) }. [ABSTRACT FROM AUTHOR]

We employ partitioning methods, in the spirit of Montiel–Ros but here recast for general actions of compact Lie groups, to prove effective lower bounds on the Morse index of certain families of closed minimal hypersurfaces in the round four-dimensional sphere, and of free boundary minimal hypersurfaces in the Euclidean four-dimensional ball. Our analysis reveals, in particular, phenomena of linear index growth for sequences of minimal hypersurfaces of fixed topological type, in strong contrast to the three-dimensional scenario. [ABSTRACT FROM AUTHOR]