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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]

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]

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 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]

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]

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, 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]

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]

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like | u | α or | u | α − 1 u (α > 1) and the initial data u 0 belongs to the Sobolev spaces L s p (G) for 1 < p < ∞ and 0 < s < N / p. Since stratified Lie groups G include the Euclidean space R n as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on R n to G. It should be noted that our proof is very different from it given by Ribaud on R n. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on R n. By using the generalized fractional chain rule on G , we can avoid the discussion of Fourier analysis on G and make the proof more simple. [ABSTRACT FROM AUTHOR]

In this work we study positivity preservers T : R [ x 1 , ... , x n ] → R [ x 1 , ... , x n ] with constant coefficients and define their generators A if they exist, i.e., exp ⁡ (A) = T. We use the theory of regular Fréchet Lie groups to show the first main result. A positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy–Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators L ̂ n with n ≥ 1 satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the β -function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map. [ABSTRACT FROM AUTHOR]

Let p and ℓ be distinct prime numbers, let q be a power of a prime number r that is distinct from p and ℓ , and let M be a positive integer coprime to q ℓ . We define the directed graph X ℓ q (M) whose vertices are given by isomorphism classes of elliptic curves over the finite field of q elements enhanced with the full level M structure. The edges of X ℓ q (M) are given by ℓ -isogenies. Fix a positive integer N, and write M = p n N . We are interested in when the connected components of X ℓ q (p n N) give rise to a tower of Galois coverings as n varies. We analyze the structure of the inverse limit of the Galois groups of these coverings as a p-adic Lie group. We also study similar towers of isogeny graphs given by oriented elliptic curves enhanced with a full level structure. [ABSTRACT FROM AUTHOR]

The present paper offers a comprehensive analysis of the one-dimensional relativistic magnetogasdynamics equations in Lagrangian coordinates. These equations, describing the behavior of a relativistic magnetized gas, are reformulated in variational form to facilitate further analysis. A complete group analysis of the resulting Euler–Lagrange equation is performed, allowing us to systematically identify the symmetries inherent in the system. Using Noether's theorem, we derive conservation laws in Lagrangian coordinates, offering critical insights into the invariants and conserved quantities of the system. This work expands the theoretical understanding of relativistic magnetogasdynamics. [ABSTRACT FROM AUTHOR]

We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected. [ABSTRACT FROM AUTHOR]

This work presents a novel approach for the optimization of dynamic systems on finite-dimensional Lie groups. We rephrase dynamic systems as so-called neural ordinary differential equations (neural ODEs), and formulate the optimization problem on Lie groups. A gradient descent optimization algorithm is presented to tackle the optimization numerically. Our algorithm is scalable, and applicable to any finite-dimensional Lie group, including matrix Lie groups. By representing the system at the Lie algebra level, we reduce the computational cost of the gradient computation. In an extensive example, optimal potential energy shaping for control of a rigid body is treated. The optimal control problem is phrased as an optimization of a neural ODE on the Lie group SE (3), and the controller is iteratively optimized. The final controller is validated on a state-regulation task. [ABSTRACT FROM AUTHOR]

We consider the variable coefficient Boiti–Leon–Pempinelli and two similar systems. We present the enhanced Lie group classification. This task is achieved with the aid of the equivalence transformations. We construct a number of similarity reductions which are used to derive exact solutions for the three (1+2)$$ \left(1&#x0002B;2\right) $$‐dimensional systems. The reductions are obtained using both one‐ and two‐dimensional subalgebras. In a specific case, each system admits a hidden symmetry. [ABSTRACT FROM AUTHOR]

We prove a nonarchimedean analogue of Jørgensen's inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$ , where K is a p -adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$ , which is a special case of a result by Breuillard and Gelander ['On dense free subgroups of Lie groups', J. Algebra 261 (2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete. [ABSTRACT FROM AUTHOR]

A celebrated result of Gromov ensures the existence of a contact structure on any connected, non-compact, odd-dimensional Lie group. In general, such structures are not invariant under left translation. The problem of finding which Lie groups admit a left-invariant contact structure resolves to the question of determining when a Lie algebra g is contact; that is, admits a one-form φ ∈ g ⁎ such that φ ∧ (d φ) k ≠ 0. In full generality, this remains an open question; however we settle it for the important category of the evocatively named seaweed algebras by showing that an index-one seaweed is contact precisely when it is quasi-reductive. Seaweeds were introduced by Dergachev and Kirillov who initiated the development of their index theory – since completed by Joseph, Panyushev, Yakimova, and Coll, among others. Recall that a contact Lie algebra has index one – but not characteristically so. Leveraging recent work of Panyushev, Baur, Moreau, Duflo, Khalgui, Torasso, Yakimova, and Ammari, who collectively classified quasi-reductive seaweeds, our equivalence yields a full classification of contact seaweeds. We remark that since type-A and type-C seaweeds are de facto quasi-reductive (by a result of Panyushev), in these types index one alone suffices to ensure the existence of a contact form. [ABSTRACT FROM AUTHOR]

This paper addresses the estimation problem of nonlinear systems evolving on Lie groups with unknown parameters. More precisely, some parameters in the equations of motion or sensor measurements are unknown, such as gravitational anomalies and measurement biases, and are infeasible to estimate with available observations. The unscented Schmidt-Kalman filter (USKF) approach in Euclidean space is incorporated with exponential maps from Lie algebra to Lie groups, to develop USKF algorithms on Lie groups. Two types of USKFs are derived, respectively, from left-invariant and right-invariant state estimation errors. The two USKFs, not only account for the effect of unknown parameters but also provide estimates preserving the geometry of state manifold. They are advantageous over the extended Schmidt-Kalman filter for nonlinear systems in the sense of avoiding the computation of Jacobian and achieving higher or comparable estimation accuracy depending on the magnitude of parameters uncertainties. The proposed method is then applied to a spacecraft attitude estimation problem based on quaternion representation, where the magnitude of the gyroscope bias noise is unknown. Simulations are conducted to illustrate the effectiveness of the proposed algorithms in comparison with other methods. [ABSTRACT FROM AUTHOR]

This paper provides extensions of the methods of Lie symmetry group and nonlinear self–adjointness to fractional differential equations involving mixed derivatives of Riemann–Liouville time-fractional derivative and first-order partial derivative. We present explicitly the general prolongation formulae expressing the action of Lie group on the mixed fractional derivatives and the expressions of conserved vectors in conservation laws. Moreover, the obtained results are used to investigate the symmetry groups and conservation laws of time-fractional Fokas–Lenells equation, whose exact solution and nontrivial conservation law are thereby constructed. [ABSTRACT FROM AUTHOR]

In this article, the Bessel and Tricomi functions are combined with Appell polynomials to introduce the families of Appell–Bessel and Appell–Tricomi functions. The 2-variable 2-parameter Hermite–Bessel and Hermite–Tricomi functions are considered as members of these families, and framed within the representation of the Lie algebra T3. Consequently, the implicit summation formulae for these functions are derived. Certain examples are also considered. The article concludes with the derivation of a relation involving the 2-variable 2-parameter Hermite–Tricomi functions by following the Weisner's approach. [ABSTRACT FROM AUTHOR]

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

In this article, we consider the equations of motion for an articulated n$$ n $$‐trailer vehicle with different masses and inertias in the presence of force and torque commands. We design a second‐order optimal filter to estimate the pose of the first vehicle and of the trailers exploiting the Lie group structure and the nonholonomic and hooking constraints. We consider the filtering problem from three different perspectives: in the first masses and inertias are time‐varying and perfectively known, in the second they are known only in the first part of the maneuver, but they are not updated over time. In the last case masses and inertias are considered within the state vector and therefore estimated by the filter. Depending on the needs in real‐life applications (e.g., whether the masses and inertias remain fixed or change during the maneuver and are unknown), the best‐performing filter can be used. The sensing system consists of a GPS‐like configuration (global positioning system) obtained by using an antenna attached to the leading car. [ABSTRACT FROM AUTHOR]

The author proceed to construct a dual pair for the AdS3×S3$AdS_3 \times S^3$ background by applying non‐Abelian T‐duality (here as Poisson–Lie [PL] T‐duality on a semi‐Abelian double). By using a certain parametrization of the 4‐dimensional Lie group A2⊗2A1${A}_2 \otimes 2{A}_1$ and by a suitable choice of spectator‐dependent matrices the original σ$\sigma$‐model including the AdS3×S3$AdS_3 \times S^3$ metric and a non‐trivial B$B$‐field are constructed. The dual background constructed by means of the PL T‐duality with the spectators is an asymptotically flat one with a potential black hole interpretation supported by a non‐trivial H$H$‐flux whose metric contains the true singularity with a single horizon. The question of classical integrability of the non‐Abelian T‐dual σ$\sigma$‐models under consideration is addressed, and their corresponding Lax pairs are found, depending on some spectral parameters. Finally, the conformal invariance conditions of the models are checked up to two‐loop order, and it has been concluded that the resulting model is indeed a solution of supergravity. [ABSTRACT FROM AUTHOR]

Wigner's classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann's theory to time-dependent phases using Hilbert bundles. [ABSTRACT FROM AUTHOR]

Due to its critical importance in engineering applications, this study is motivated by the essential need to understand natural convection over a vertical cylinder with combined heat and mass transfer. Lie group symmetry transformations are used to analyze the thermal and velocity boundary layers of steady, naturally convective laminar fluid flow over the surface of a vertical cylinder. The one-parameter Lie group symmetry technique converts the system of governing equations into ordinary differential equations, which are then solved numerically using the implicit Runge–Kutta method. The effect of the Prandtl number, Schmidt number, and combined buoyancy ratio parameter on axial velocity, temperature, and concentration profiles are illustrated graphically. A specific range of parameter values was chosen to compare the obtained results with previous studies, demonstrating the accuracy of this method relative to others. The average Nusselt number and average Sherwood number are computed for various values of the Prandtl number Pr and Schmidt number Sc and presented in tables. It was found that the time required to reach a steady state for velocity and concentration profiles decreases as the Schmidt number Sc increases. Additionally, both temperature and concentration profiles decrease with an increase in the combined buoyancy ratio parameter N. Flow reversal and temperature defect with varying Prandtl numbers are also shown and discussed in detail. [ABSTRACT FROM AUTHOR]

In this article, the propagation of strong converging cylindrically symmetric shock waves in ideal dusty gas is studied using the Lie group technique while considering the effect of an axial magnetic field in a rotating gas atmosphere. The constant density in an undisturbed medium is assumed, whereas the magnetic field, the azimuthal, and axial components of fluid velocity are considered to be varying. The arbitrary constants appearing in the expressions for infinitesimals of the Local Lie group of transformations bring about three different cases of solutions, i.e., with power-law shock path, exponential-law shock path, and a particular case of power-law shock path. Numerical solutions are obtained in the cases of the power-law shock path. The self-similar solutions to the problem are obtained, and the effect of the Shock Cowling number, the mass concentration of solid dust particles, the relative specific heat, the ratio of the density of solid particles, and the ambient azimuthal velocity exponent on the shock evolution are depicted through graphs. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Hyperstructures is the property of University of Mohaghegh Ardabili and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Finsler Geometry & its Applications is the property of University of Mohaghegh Ardabili and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity. [ABSTRACT FROM AUTHOR]

In the present article we study the periodic structure of some well-known classes of C 1 self-maps on the product of spheres of different dimensions: transversal maps, Morse-Smale diffeomorphisms and maps with all its periodic points hyperbolic. Our approach is via the Lefschetz fixed point theory. We give a complete characterization of the minimal set of Lefschetz periods for Morse-Smale diffeomorphisms on these spaces. We also consider C 1 maps with all its periodic points hyperbolic and we give conditions for these maps to have infinitely many periodic points. We describe the period set of the transversal maps on these spaces. Finally we applied these results to describe the periodic structure of similar classes of maps on compact, connected, simply connected and simple Lie groups. [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