Search Results (436)

Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which is obtained from a general time-dependent quadratic Hamiltonian. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three conserved quantities, which are time-dependent quadratic forms in position and momentum, and trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p are defined. It shows that, if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables is obtained as i, and in the case of monotonic motion, it is obtained as 1. Full article

This paper studies a realization-theoretic problem inside the standard Lorentz-covariant Fourier-dual framework on $L^2\left({\mathbb{R}}^{3,1}\right)$: whether position-space and momentum-space geometric translations can be placed on equal structural footing without leaving the ordinary X- and K-polarized realizations. Working on the common Schwartz core $\mathcal{S}\left({\mathbb{R}}^{3,1}\right)$, we first isolate a Fourier-compatibility obstruction: Fourier transform exchanges geometric translations with character actions, while the Poincaré algebra contains at most one Lorentz-covariant abelian translation ideal. The main result is that, within the resulting Fourier-compatible realization class, the minimal operator-generated Lie algebra is the Lorentz–Heisenberg algebra. We then determine the full center of its universal enveloping algebra, derive the normalized Lorentz-bivector invariants, orbit data, and connected stabilizers in nondegenerate sectors, and show that the orbit variable is a normalized Lorentz bivector rather than a momentum vector. Finally, for fixed spectral elements in the dual translation sectors, we derive the associated scalar, Dirac, and vector equations in position and momentum space and show that, in the regular polarized realizations, the represented Heisenberg sector induces dual local abelian phase groups, compatible covariant derivatives, curvatures, and primary Dirac–Maxwell systems. Full article

Based on the Lie group method (symmetry transformation groups), an analysis of an unsteady (start-up) flow over a flat surface was performed. This approach enabled reducing the number of independent arguments, which significantly simplifies the process of numerical modeling. An unsteady solution was obtained for the velocity profile in the boundary layer. This enabled estimating the dynamics of the velocity profile transformation and its transition to a steady-state mode. It was shown that in the limit of infinite time of the process, the velocity profile tends to the classical steady-state Blasius profile in the boundary layer. The dynamics of the friction coefficient variation over time were elucidated too. Full article

The (2+1)-D generalized B-type Kadomtsev–Petviashvili (BKP) equation is studied in this work utilizing Painlevé property, Lie-symmetry method and generalized Kudryashov method (GKM). This study aims to pass the Painlevé test and obtain variant exact solutions for the (2+1)-D BKP equation that occurs in physical dynamics. First, we demonstrated that the governing equation exceeds the Painlevé test by using the Painlevé property. Symmetry analysis is utilized to obtain infinitesimals and vector fields of the BKP equation. The governing equation was converted to several ordinary differential equations (ODEs) using linear combinations of these vectors. GKM is used to generate a novel class of closed-form solutions for the BKP equation. Many random constants and functions were included in the derived solutions to improve their dynamic characteristics. The emergence of solutions was facilitated by the optimal selection of estimates for these elective constants. There are several types of solution behavior, such as a kink wave, solitary wave, anti-kink wave, and single wave. Full article

The $(2+1)$-dimensional Boussinesq equation plays an important role in mathematical physics. In this paper, we investigate some exact solutions of the $(2+1)$-dimensional variable-coefficient Boussinesq equation. Firstly, the Painlevé analysis is carried out, and an auto-Bäcklund transformation is constructed by means of a truncated Painlevé expansion combined with symbolic computation. Then, a class of new soliton-type solutions is derived. By selecting appropriate parameter values, detailed simulations are presented to illustrate the dynamical behavior of water wave propagation. Finally, the Lie point symmetries of the equation are studied, and several similarity reductions are derived by solving the corresponding characteristic equations. Full article

Deep learning systems deployed in regulated settings require explanations that are accurate and stable under nuisance transformations, yet classical post hoc transition matrices rely on fidelity-only fitting that fails to guarantee consistent explanations under spatial rotations or other group actions. In this work, we propose Equivariant Transition Matrices, a post hoc approach that augments transition matrices with Lie-group-aware structural constraints to bridge this research gap. Our method estimates infinitesimal generators in the formal and mental feature spaces, enforces an approximate intertwining relation at the Lie algebra level, and solves the resulting convex Least-Squares problem via singular value decomposition for small networks or implicit operators for large systems. We introduce diagnostics for symmetry validation and an unsupervised strategy for regularization weight selection. On a controlled synthetic benchmark, our approach reduces the symmetry defect from 13,100 to 0.0425 while increasing the mean squared error marginally from 0.00367 to 0.00524. On the MNIST dataset, the symmetry defect decreases by 72.6 percent (141.19 to 38.65) with changes in structural similarity and peak signal-to-noise ratio below 0.03 percent and 0.06 percent, respectively. These results demonstrate that explanation-level equivariance can be reliably imposed post-training, providing geometrically consistent interpretations for fixed deep models. Full article

Let $\mathcal{N}$ be a nest on a Banach space, and let $Alg\left(\mathcal{N}\right)$ denote the associated nest algebra equipped with the operator norm. In this paper, we develop a Banach–Lie framework for bounded triple derivations and triple automorphisms on $Alg\left(\mathcal{N}\right)$. We prove that the space of bounded triple derivations is closed under the commutator bracket and hence forms a Lie algebra, while the set of triple automorphisms forms a norm-closed subgroup of $\mathrm{GL}(Alg(\mathcal{N}\left)\right)$. We further establish an exponential–differential correspondence between these two classes: the exponential of a bounded triple derivation yields a one-parameter group of triple automorphisms, and conversely, the tangent space at the identity of the triple automorphism group is identified with the Lie algebra of bounded triple derivations. We also relate these objects to derivations and automorphisms of the standard embedding Lie algebra associated with the Lie triple system naturally induced by $Alg\left(\mathcal{N}\right)$. To illustrate the general theory, we finally determine explicit forms of triple derivations and triple automorphisms for the eight non-perfect three-dimensional real Lie algebras. Full article

This study aims to advance the understanding of laminar magnetohydrodynamic (MHD) fluid flow between two parallel plates subjected to a uniform transverse magnetic field, motivated by its significant applications in engineering and industrial systems such as nuclear reactor cooling, MHD generators, and electromagnetic pumping devices. The governing equations are simplified using a one-parameter Lie group symmetry transformation, which exploits the inherent symmetry properties of the system to reduce the original unsteady partial differential equations to a system of ordinary differential equations. The reduced equations are solved exactly under appropriate boundary and initial conditions, ensuring mathematically consistent and physically realistic solutions. A comprehensive analysis is conducted to examine the influence of key physical parameters, along with the applied magnetic field, on the velocity, temperature, and concentration profiles. The selected parameter ranges encompass a broad spectrum of physically relevant cases, enabling a detailed assessment of their effects. The results indicate that the transverse magnetic field exerts a damping effect on the flow, reducing the velocity profile due to the Lorentz force. Moreover, an increase in the Schmidt number accelerates the achievement of a steady-state concentration, while higher Prandtl numbers reduce the temperature profile. In contrast, the radiation parameter enhances the temperature distribution. In addition, the skin-friction coefficient is presented graphically, and the Nusselt number is evaluated to characterize the heat transfer performance. Overall, the findings provide valuable insight into the effects of magnetic, thermal, and solutal parameters on laminar MHD flow between parallel plates. Full article

Starting from the Noether first theorem, we discuss a criterion for identifying physically consistent gravitational models. In particular, we demonstrate that applying the Lie derivative to a point-like Lagrangian makes it possible to determine the underlying symmetries, their associated generators, and the corresponding conserved quantities. This method is then generalized through its first prolongation and applied to several point-like Lagrangians, with special attention to $f\left(R\right)$ gravity in a cosmological setting. In each example, the existence of symmetries results in a simplification of the dynamical system, enabling the integration of the equations of motion and the derivation of exact solutions. It is worth noticing that the existence of symmetries is always related to physically consistent models. Full article

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces $\mathrm{U}/\mathrm{H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U}/\mathrm{H}$ provided by generalized thermodynamics à la Souriau. Our main result is the proof that $\mathrm{U}/\mathrm{H}$.s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset \mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset \mathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi–Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao’s, Chentsov’s, and Amari’s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against $\mathrm{U}$ transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$. Full article

The full symmetry group is found for a system of nonlinear schrödinger equations describing the propagation of optical pulses in an isotropic media. It is shown, in particular, that the six-dimensional symmetry group found is composed of a scaling transformation and a rotation of the four-dimensional space, thereby proving that the symmetry group preserves the shape of solutions. A symmetry classification of one-dimensional subalgebras of the Lie algebra is performed and yields, in particular, the symmetry reduction to the most general system of equations satisfied by the solitary waves of the equation. Explicit soliton solutions of the equation are found by largely autonomous technics. The found solitons are used to recursively generate two new ones by means of two iterations using the symmetry group. Other properties of the system are also highlighted, as well as the possible connections between the theories of symmetry groups and Darboux transformations inspired by this study. Full article

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ${\iota }_{\mathcal{X}}\omega$. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group ${\mathrm{Aff}}^{+}\left(1\right)$, this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group $H_3$, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups. Full article

The Hodgkin–Huxley equations have successfully described neuronal excitability for over seventy years, yet their mathematical structure remains empirically justified rather than theoretically explained. Why are gating variables bounded between 0 and 1? Why does sodium conductance depend on m³h rather than other combinations? Why does potassium depend on n⁴? Why do all rate functions contain exponential voltage dependencies? Why are the kinetics first-order? We demonstrate that these structural features arise naturally from three fundamental physical symmetries governing ion channel dynamics: the compactness of conformational state space, the scaling invariance of membrane conductance, and temporal translation invariance. Using Lie group theory, we show that these symmetries uniquely determine a mathematical structure in which: (1) gating variables are necessarily bounded, (2) voltage dependencies must be exponential, (3) exponents must be integers, and (4) kinetics must be first-order. The Hodgkin–Huxley equations, rather than mere empirical fits, emerge from fundamental symmetry principles. This framework establishes that neural electrophysiology obeys the same theoretical principles as modern physics, where symmetries constrain the form of dynamical equations. It further provides a principled basis for interpreting deviations from classical behavior as manifestations of additional symmetries or symmetry breaking. Full article

Dissipative systems that evolve on time-dependent domains occur across systems science whenever redistribution, loss, and global restructuring are coupled with geometric change. This work develops a phenomenological, system-level framework for analyzing such processes and focuses on invariant organizational constraints rather than on microscopic mechanisms or specific physical realizations. Redistribution on an evolving domain is modeled through a diffusion–dissipation equation with curvature- and volume-dependent dissipative loss terms, interpreted as effective drivers of irreversible reorganization. Lie symmetry analysis reveals a non-semisimple structure whose generators act as invariants of admissible system-level reorganizations rather than as sources of conservation laws. By selecting a symmetry-compatible subalgebra, an emergent geometric representation is constructed that compactly encodes global balance constraints without invoking a physical spacetime interpretation. The framework yields time-independent geometric invariants and a system-level balance relation that stabilizes global organization despite ongoing local dissipation. A dimensionless geometric indicator is introduced to quantify intrinsic anisotropy of reorganization and to classify dissipative regimes. Owing to its invariant and phenomenological character, the approach is applicable to a broad class of complex systems with evolving domains and irreversible dynamics, consistent with the scope of systems research. Full article

The (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) problem was examined in this paper using the developed Exp-function method (DEFM) and Lie symmetry analysis. The objective of this research is studying the BKP equation to get novel exact solutions. Symmetry analysis has been used to determine similarity variables and vector fields. The governing equation was reduced to five variant ordinary differential equations (ODEs). The DEFM was employed for four of them to obtain several novel exact solutions that contain arbitrary constants. The most appropriate choice of values for these optional constants contributed to the emergence of solutions, such as double waves, multisolitons, kink waves, anti-kink waves, and solitary waves. The obtained exact solutions are presented in a 3D graph. The behavior of the solutions can be utilized to explore the application of the governing equation in fluid dynamics, plasma physics, nonlinear optics, and ocean physics. Full article