Lie symmetry analysis and invariant solutions for the(3+1)-dimensional Virasoro integrable model

Lie symmetry analysis is applied to a(3+1)-dimensional Virasoro integrable model and the corresponding similarity reduction equations are obtained with the different infinitesimal generators.Invariant solutions with arbitrary functions for the(3+1)-dimensional Virasoro integrable model,including the interaction solution between a kink and a soliton,the lump-type solution and periodic solutions,have been studied analytically and graphically.

POINCARE—CARTAN INTEGRAL INVARIANTS OF BIRKHOFFIAN SYSTEMS

Based on modern differential geometry, the symplectic structure of a Birkhoffiansystem which is an extension of conservative and nonconservative systems is analyz ed. Anone-dimensional damped vibration is taken as an ilhustrative example and an integralinvariant of Poincaré's type is found.

From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator,the second class of integral theorems,the Ganssian-curvature-based integral theorems and the Gaussian (or spheri- cal)mapping,a series of invariants or geometric conservation quantities under Gaussian (or spherical)mapping are revealed.From these mapping invariants important trans- formations between original curved surface and the spherical surface are derived.The potential applications of these invariants and transformations to geometry are discussed.

A Note on Calculation of Phase Space Path Integral by Means of Invariants

在这笔记，我们使用方法对不变计算出现在我们的以前的文章的阶段空间路径积分。

INVARIANT FORM AND INTEGRAL INVARIANTS ON KAHLER MANIFOLD

The important notions and results of the integral invariants of Poincare and Cartan-Poincare and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kahler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider and deeper results.

Mathematical Aspects of SU (2) and SO(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

ABOUT THE BASIC INTEGRAL VARIANTS OF HOLONOMIC NONCONSERVATIVE DYNAMICAL SYSTEMS

In this paper,we prove that for holonomic nonconservative dynamical system the Poincareand Poincaré-Cartan integral invariants do not exist.Instead of them,we introduce the integral variants ofPoincaré Cartan’s type and of Poincaré’s type for holonomie noneonservative dynamical systems,and usethese variants to solve the problem of nonlinear vibration.We also prove that the integral invariants intro-duced in references[1]and[2]are merely the basic integral variants given by this paper.

Isotropic polynomial invariants of Hall tensor

The Hall tensor emerges from the study of the Hall effect, an important magnetic effect observed in electric conductors and semiconductors. The Hall tensor is third-order and three-dimensional, whose first two indices are skew-symmetric. This paper investigates the isotropic polynomial invariants of the Hall tensor by connecting it with a second-order tensor via the third-order Levi-Civita tensor. A minimal isotropic integrity basis with 10 invariants for the Hall tensor is proposed. Furthermore, it is proved that this minimal integrity basis is also an irreducible isotropic function basis of the Hall tensor.

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.

A NECESSARY CONDITION FOR CERTAIN INTEGRAL EQUATIONS WITH NEGATIVE EXPONENTS

This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a pair of positive Lebesgue measurable solutions of this integral system, then 1/(p-1)+1/(q-1)=λ/n, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.

Remove Bertrand＇s Paradox by Means of Invariant Measure

By means of invariant measure, Bertrand's paradox was removed.

Application of Differential Invariant Method for Solving the Electromagnetic Fields

We study the first integral and the solution of electromagnetic field by Lie symmetry technique and the differential invariant method.The definition and properties of differential invariants are introduced and the infinitesimal generators of Lie symmetries and the differential invariants of electromagnetic field are obtained.The first integral and the solution of electromagnetic field are given by the Lie symmetry technique and the differential invariants method.A typical example is presented to illustrate the application of our theoretical results.

Weighted Variation Inequalities for Singular Integrals and Commutators in Rearrangement Invariant Banach and Quasi-Banach Spaces

In order to study the boundedness of some operators in general function spaces which include Lorentz spaces and Orlicz spaces as special examples,Lorentz introduced a new space called rearrangement invariant Banach function spaces,denoted by RIBFS.It is shown in this paper that variation operators of singular integrals and their commutators are bounded on RIBFS whenever the kernels satisfy the L~r-H?rmander conditions.Moreover,we obtain some quantitative weighted bounds in the quasi-Banach spaces and modular inequalities for above variation operators and their commutators.

Hamiltonian, Path Integral and BRST Formulations of the Restricted Gauge Theory of <i>QCD₂</i>

We study the Hamiltonian, path integral and Becchi-Rouet-Stora and Tyutin (BRST) formulations of the restricted gauge theory of QCD2 à la Cho et al. under appropriate gauge-fixing conditions.

Generalized Noether Theorem and Poincaré-Cartan Integral Invariant for Singular High-order Lagrangian in Fields Theories

A generalized first Noether theorem (GFNT) originating from the invariance under the finite continuous group for singular high-order Lagrangian and a generalized second Noether theorem (or generalized Noether identities (GNI)) for variant system under the infinite continuous group of field theory in canonical formalism are derived. The strong and weak conservation laws in canonical formalism are also obtained. It is pointed out that some variant systems also have Dirac constraint. Based on the canonical action, the generalized Poincaré-Cartan integral invariant (GPCⅡ) for singular high-order Lagrangian in the field theory is deduced. Some confusions in literafure are clarified. The GPCⅡ connected with canonical equations and canonical transformation are discussed.

First integral and integral invariant of Birkhoffian system

The variational equation of the Birkhoffian system is established, by which it is proved that an integral invariant can be constructed with a known first integral. And its inverse is also correct.

Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)

In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degreesof invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that anyholomorphic foliation of degree 2 does not have cubic limit cycles.

High-Dimensional Schwarzian Derivatives and Painlevé Integrable Models

Because all the known integrable models possess Schwarzian forms with Mobious transformation invariance,it may be one of the best ways to find new integrable models starting from some suitable Mobious transformation invariant equations. In this paper, we study the Painlevé integrability of some special (3+1)-dimensional Schwarzian models.

The Integrable Conditions of Riccati Differential Equation

In this paper,the new integrable conditions of Riccati equation is presented by invarant of Riccati equation.

FIRST INTEGRAL AND INTEGRAL-INVARIANT FOR NONHOLONOMIC SYSTEMS

Ⅰ. INTRODUCTION The first integrals and the integral-invariants of mechanical systems are important contents of the integral theory of the equations of motion, between which there exists close