Deformation of the exterior algebra Ω(M n) and the Yang-Baxter equation

We show that the deformation of the exterior algebra on a given manifold is related to the existence of the Yang-Baxter equation. We prove that this deformed algebra involves a differential operator generating the algebra. The obtained differential calculus is not commutative and we recover the classical one for the classical limit of the deformation parameters. The q-analogue of the Leibniz rule corresponding to the purposed differential operator is given.     

Using the Exterior Matrix Method to find eigenfrequencies when the governing equations are systems of first-order equations

In this paper, we will apply the Exterior Matrix Method to the case where the governing equations for one piece of the structure is a system of 4 first-order equations. This generalizes a result when the governing equation was a single fourth-order equation. Ironically, the governing equations do not have to be solved in order to find the corresponding exterior matrix, and the exterior matrices can be used to find the eigenfrequencies of the system, even if there are dissipative joints added to the system. We will first look at the well-understood example of Euler–Bernoulli beams to illustrate the concept, and then move on to the more difficult inclined cable problem.     

On the covariance of the Fokker-Planck equation

We study the covariance of the Fokker-Planck equation under general gross variables transformations by means of Riemann differential geometry using an affine connection Γ^?_μν unsymmetric in their lower indices and without assuming that the covariant derivative of the diffusion tensor D^μν_;? be zero. We come to the conclusion that to achieve our aim we only need the value of the contraction Γ^?_?ν, all other components of the affine connection remaining completely arbitrary. We argue, therefore, that the most economic way of presenting the covariance of the Fokker-Planck equation is by means of exterior differential calculus. As an application we study physical systems under detailed balance showing that for them the irreversible part of the contravariant drift vector, that is then uniquely determined, is related only to a symmetric tensor while its reversible component is exclusively related to an antisymmetric tensor. A criticism of a compact Fokker-Planck equation is also included.     

A Self-Similar Dynamics in Viscous Spheres

We study the evolution of radiating and viscous fluid spheres assuming an additional homothetic symmetry on the spherically symmetric space-time. We match a very simple solution to the symmetry equations with the exterior one (Vaidya). We then obtain a system of two ordinary differential equations which rule the dynamics, and find a self-similar collapse which is shear-free and with a barotropic equation of state. Considering a huge set of initial self-similar dynamics states, we work out a model with an acceptable physical behavior.     

The Dirac equation in exterior form

Using the correspondence between the Clifford and exterior algebras we write the Dirac equation in terms of differential forms. The covariances of the theory are then examined. We show in detail the correspondence with usual matrix methods.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

Stochastic Cohomology of the Frame Bundle of the Loop Space

Abstract

We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose some regularity assumptions over the kernels of the differential forms. This allows us to define an exterior stochastic differential derivative over these forms.     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

A note on the Lie symmetries of complex partial differential equations and their split real systems

Folklore suggests that the split Lie-like operators of a complex partial differential equation are symmetries of the split system of real partial differential equations. However, this is not the case generally. We illustrate this by using the complex heat equation, wave equation with dissipation, the nonlinear Burgers equation and nonlinear KdV equations. We split the Lie symmetries of a complex partial differential equation in the real domain and obtain real Lie-like operators. Further, the complex partial differential equation is split into two coupled or uncoupled real partial differential equations which constitute a system of two equations for two real functions of two real variables. The Lie symmetries of this system are constructed by the classical Lie approach. We compare these Lie symmetries with the split Lie-like operators of the given complex partial differential equation for the examples considered. We conclude that the split Lie-like operators of complex partial differential equations are not in general symmetries of the split system of real partial differential equations. We prove a proposition that gives the criteria when the Lie-like operators are symmetries of the split system.     

Geometrical formulation of equilibrium phenomenological thermodynamics

An abstract geometric formulation of equilibrium phenomenological thermodynamics which generalizes and unifies that of Gibbs, Carathéodory, and others is given. As done by Hermann, one adapts here a contact manifold as the basic mathematical structure. Such a manifold for a thermodynamic system is constructed. The empirical laws of thermodynamics have been reformulated in terms of this manifold and by means of exterior differential forms. Such concepts as Gibbs phase rule, Gibbs-Duhem equation and thermodynamic potentials have been reexamined within such a general scheme.     

An asymptotic expansion for the Neumann sieve problem

In this paper, we study the Neumann sieve problem for the Laplace equation. Our objective is to compute the complete asymptotic expansion for the problem. The expansion consists of the interior part, in the vicinity of the filter, and an exterior part, far away from the filter. The interior approximation is a Bakhvalov-Panasenko-type expansion with terms defined by a sequence of auxiliary problems on infinite stripes and matching with the exterior expansion. We prove the related error estimate.     

An improved singular perturbation solution for bound geodesic orbits in the Schwarzschild metric

A modification to the Lindstedt-Poincaré method of strained parameters is applied to the differential equation of the orbit of a test particle in the Schwarzschild exterior metric. A new perturbation solution for the equation of the bound orbit, which is completely free of secular terms in the angular coordinate, is derived. The precession of the orbit per revolution is calculated using this solution and it is found to give a more accurate result than existing perturbation solutions. The method should be applicable to similar orbital problems in general relativity.     

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders,Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation.We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure.The method can be generalized to the other fractional soliton hierarchy.     

The relativistic Riemann invariants and the propagation of initial data surfaces

A method is described by which the relativistic Riemann invariants can be found for a fluid with an arbitrary equation of state, undergoing dissipation and moving in a general metric. Specific formulae are derived for a spherically symmetric system. Limiting cases defined by relativistic and non-relativistic gases, both warm, cold, fast and slow are examined. We prove that the invariants do exist, and a necessary and sufficient condition for their determination is the solution of a differential equation with the structure of an exterior one form of two components. The common parameter of these components is the characteristic space-time direction which is also derived in the process of determining the invariants. The characteristic surfaces, being the surfaces over which initial data is carried, all coalesce to the forward light cone in the extreme relativistic limit. Relativistic fluids emanating from receding sources appear to increase their internal kinetic energy as they decelerate.A non-linear distance-velocity relation for these waves is evident in the differential equations which are found. Their full meaning remains to be explored.     

A path integral method for coarse-graining noise in stochastic differential equations with multiple time scales

We present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales. The objective of this method is to derive from the original system a new stochastic differential equation describing the system’s evolution on slow time scales. For this purpose, we start from the corresponding path integral representation of the stochastic system and apply a multi-scale expansion to the associated path integral kernel of the corresponding Lagrangian. As a concrete example, we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion-managed fiber-optic communications. Moreover, we show that, for this particular example, the new path integration method yields the same result at leading order as an asymptotic expansion of the associated Fokker-Planck equation.     

The relativistic Riemann invariants and the propagation of initial data surfaces

A method is described by which the relativistic Riemann invariants can be found for a fluid with an arbitrary equation of state, undergoing dissipation and moving in a general metric. Specific formulae are derived for a spherically symmetric system. Limiting cases defined by relativistic and non-relativistic gases, both warm, cold, fast and slow are examined. We prove that the invariants do exist, and a necessary and sufficient condition for their determination is the solution of a differential equation with the structure of an exterior one form of two components. The common parameter of these components is the characteristic space-time direction which is also derived in the process of determining the invariants. The characteristic surfaces, being the surfaces over which initial data is carried, all coalesce to the forward light cone in the extreme relativistic limit. Relativistic fluids emanating from receding sources appear to increase their internal kinetic energy as they decelerate. A non-linear distance-velocity relation for these waves is evident in the differential equations which are found. Their full meaning remains to be explored.     

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.     

On a scale function for testing the conformality of a Finsler manifold to a Berwald manifold

In this paper, we are going to discuss the problem whether how we can check the conformality of a Finsler manifold to a Berwald manifold. The method is based on a differential 1-form constructing on the underlying manifold by the help of integral formulas such that its exterior derivative is conformally invariant. If the Finsler manifold is conformal to a Berwald manifold, then the exterior derivative vanishes. This gives the following necessary condition: the differential form is closed and, at least locally, it is exact as the exterior derivative of a scale function for testing the conformality. A necessary and sufficient condition is also given in terms of a distinguished linear connection on the underlying manifold – it is expressed by the help of canonical data. In order to illustrate how we can simplify the process in special cases Randers manifolds are considered with some explicit calculations.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.