We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young ineq...
We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical an...
Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic spa...
We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.
Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the s...
In this short note, we prove two properties of symplectic pairs on a four-manifold: firstly we prove that two transversal orientable foliations of codimension two, which are taut for the same Riemannian metric, are the characteristic foliations of a...
In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the can...
We study new geometrical and topological aspects of polarized k-symplectic manifolds. In addition, we study the De Rham cohomology groups of the k-symplectic group. In this work, we pay particular attention to the problem of the orientation of polari...
Real systems are often complex, nonlinear, and noisy in various fields, including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonline...
Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reductio...
The symmetry of the spectrum and the completeness of the eigenfunction system of the Hamiltonian operator matrix have important applications in the symplectic Fourier expansion method in elasticity. However, the symplectic self-adjointness of Hamilto...
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum system...
The multi-symplectic integrator, as a numerical integration approach with symmetry, is known to have the characteristic of preserving the qualitative features and geometric properties of certain systems. Using the multi-symplectic integrator, the num...
The all-at-once technique has attracted many researchers’ interest in recent years. In this paper, we combine this technique with a classical symplectic and symmetric method for solving Hamiltonian systems. The solutions at all time steps are obtaine...
Let G be a compact connected Lie group, (X,ω,μ) a Hamiltonian G-manifold with moment map μ, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices (A,u) near Z, where A is...
In the previous work, the author gave the following symplectic/contact geometric description of the Bayesian inference of normal means: The space H of normal distributions is an upper halfplane which admits two operations, namely, the convoluti...
The symplectic method for a thin piezoelectric plate problem is developed. The Hamiltonian canonical equation of thin piezoelectric plate is given by using the variational principle. By applying the separation of variables method, we can obtain sympl...
We summarise recent work on the classical result of Kirillov that any simply connected homogeneous symplectic space of a connected group G is a hamiltonian G^-space for a one-dimensional central extension G^ of G, and is thus (by a result of Kostant...
Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it...
We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of “quantum blobs” introduced in previous work; quantum blobs are the sm...
Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-p...
In this article, we construct a kind of Maslov-type index for general paths of 2×2 symplectic matrices that have two arbitrary endpoints. Our method is consistent and direct no matter whether the starting point of the path is an identity or not...
In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t-dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations in...
A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Rung...
Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of th...
Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in th...
We take into account the dynamics of three types of models of rotating galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric potential perturbations, the angular momentum varies with time, and the kinetic energy depends on the mo...
The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geome...
Poincaré’s Recurrence Theorem implies that any isolated Hamiltonian system evolving in a bounded Universe returns infinitely many times arbitrarily close to its initial phase space configuration. We discuss this and related recurrence pr...
We study the complete, compact, locally affine manifolds equipped with a k-symplectic structure, which are the quotients of Rn(k+1) by a subgroup Γ of the affine group A(n(k+1)) of Rn(k+1) acting freely and properly discontinuously on Rn(k+1) and lea...
In recent works by Wu and Wang a class of explicit symplectic integrators in curved spacetimes was presented. Different splitting forms or appropriate choices of time-transformed Hamiltonians are determined based on specific Hamiltonian problems. As...
A non-autonomous memristor circuit based on van der Pol oscillator with double periodically forcing term is presented and discussed. Firstly, the differences of the van der Pol oscillation of memristor model between Euler method and symplectic Euler...
We introduce quasi self-dual (QSD), linear complementary dual (LCD), and additive complementary dual (ACD) codes for the symplectic inner product over a non-commutative non-unitary ring of order 9. We establish connections with symplectic–self-...
To date, all explicit symplectic Runge–Kutta–Nyström methods of order five or above are derived by numerical solutions of order condition equations and symplectic condition. In this paper, we derive 124 sets of seven-stage fifth-orde...
The rapid integration of renewable energy sources into modern power systems has exacerbated power quality challenges, particularly broadband oscillation phenomena that threaten grid symmetry and stability. The proposed symplectic geometric mode decom...
This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of t...
The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynam...
In this paper, we prove that the complex four dimensional compact holomorphic symplectic manifold we found earlier is not formal. This gives another strong consequence that it is not a topological Kähler manifold. We also conjecture that this is...
The non-stationary characteristics of the vibration signals of rolling bearings will be aggravated under variable speed conditions. Meanwhile, multichannel signals can provide a more comprehensive characterization of state information, providing mult...
A fourth-order energy preserving composition scheme for multi-symplectic structure partial differential equations have been proposed. The accuracy and energy conservation properties of the new scheme were verified. The new scheme is applied to solve...
A strong Gelfand pair (G,H) is a finite group G together with a subgroup H such that every irreducible character of H induces to a multiplicity-free character of G. We classify the strong Gelfand pairs of the symplectic groups Sp4(q) for even q.
A brane-world metric with an external magnetic field is a modified theory of gravity. It is suitable for the description of compact sources on the brane such as stars and black holes. We design a class of explicit symplectic integrators for this spac...
This paper aims to explore the difficulty of obtaining source signals from complex mixed signals and the issue that the FastICA algorithm cannot directly decompose the received single-channel mixed signals and distort the signal separation in low sig...
In multimodal collaborative learning, the gradient dynamics of heterogeneous modalities face significant challenges due to the curvature heterogeneity of parameter manifolds and mismatches in phase evolution. Traditional Euclidean optimization method...
Bearing vibration response studies are crucial for the condition monitoring of bearings and the quality inspection of rotating machinery systems. However, it is still very difficult to diagnose bearing faults, especially rolling element faults, due t...
We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity mome...
Stability and convergence analyses of the multi-symplectic variational integrator for the nonlinear Schro¨dinger equation are discussed in this paper. The variational integrator is proved to be unconditionally linearly stable using the von Neuman...
Aiming at fault feature extraction of a hydraulic pump signal, a new method based on symplectic geometry mode decomposition (SGMD) and power spectral entropy (PSE) is proposed. First, the SGMD is applied to decompose a multi-component fault signal, t...
In this paper, a novel analytical approach for the buckling of ring-stiffened porous graphene platelet-reinforced composite cylindrical shells under hydrostatic pressure is proposed under the framework of symplectic mechanics. Three types of graphene...
Electrical networks face operational challenges from power quality-affecting disturbances. Since disturbance signatures directly affect classifier performance, optimized feature selection becomes critical for accurate power quality assessment. The pu...
of 5