Spatiotemporal and statistical symmetries

The notion of symmetries, either statistical or deterministic, can be useful for the characterization of complex systems and their bifurcations. In this paper, we investigate the connection between the (microscopic) spatiotemporal symmetries of a space-time functionu(x, t), on the one hand, and the (macroscopic) symmetries of statistical quantities such as the spatial (resp. temporal) two-point correlations and the spatial (resp. temporal) average, on the other hand. We show, how, under certain conditions, these symmetries are related to the symmetries of the orbits described byu(x, t) in the characteristic (phase) spaces. We also determine the largest group of spatiotemporal symmetries (in the sense introduced in our earlier work) satisfied by a given space-time functionu(x, t) and indicate how to extract the subgroups of point symmetries, namely those directly implemented on the space and time variables. Conversely, we determine all the functions invariant by a given space-time symmetry group. Finally, we illustrate all the previous points with specific examples.     

Differential Equations Invariant Under Conditional Symmetries

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous Korteweg–de Vries like equations are given to show and clarify the methodology introduced.     

Pseudo-Hermitian Hamiltonians and Generalized Involutory Symmetries

We propose definitions of generalized parity (P), time-reversal (T) and charge-conjugation (C) operators such, that any diagonalizable pseudo-Hermitian Hamiltonian is invariant under the involutory symmetries C, TP, and CPT. We inquire about the peculiarities of such symmetries showing that these constitute the P-unitary and P-antiunitary symmetry generators. Moreover, we give a necessary and sufficient condition for diagonalizable pseudo-Hermitian Hamiltonians to admit P-pseudounitary and P-pseudoantiunitary symmetries.     

2+1维广义Calogero-Bogoyavlenskii-Schiff方程的无穷多对称及其约化

通过潘勒卫检验,得到了2+1维广义Calogero-Bogoyavlenskii-Schiff方程可积的条件.在这个基础上,得到了GCBS方程的双线性形式,从而根据形式级数展开法得到了无穷多对称.根据这个对称可以得到GCBS方程的约化. 关键词： 无穷多对称 截断对称 对称约化 GCBS方程     

Kac-Moody-Virasoro symmetry algebra of a （2＋1）-dimensional bilinear system

Based on some known facts of integrable models, this paper proposes a new （2＋1）-dimensional bilinear model equation. By virtue of the formal series symmetry approach, the new model is proved to be integrable because of the existence of the higher order symmetries. The Lie point symmetries of the model constitute an infinite dimensional Kac- Moody Virasoro symmetry algebra. Making use of the infinite Lie point symmetries, the possible symmetry reductions of the model are also studied     

Extended antifield formalism

The antifield formalism is extended so as to incorporate the rigid symmetries of a given theory. To that end, it is necessary to introduce global ghosts not only for the given rigid symmetries, but also for all the higher order conservation laws, associated with conserved antisymmetric tensors j^μ₁3μ_k fulfilling _μ1j^μ₁3μ_k 2˜ 0. Otherwise, one may encounter obstructions of the type discussed in by the authors. These higher order conservation laws are shown to define additional rigid symmetries of the master equation and to form — together with the standard symmetries — an interesting algebraic structure. They lead furthermore to independent Ward identities which are derived in the standard manner, because the resulting master (“Zinn-Justin”) equation capturing both the gauge symmetries and the rigid symmetries of all orders takes a known form. Issues such as anomalies or consistent deformations of the action preserving some set of rigid symmetries can be also systematically analysed in this framework.     

均匀磁场中二维各向同性带电谐振子的守恒量与对称性研究

由牛顿第二定律得到二维各向同性带电谐振子在均匀磁场中运动的运动微分方程,通过对运动微分方程的直接积分得到系统的两个积分(守恒量).利用Legendre变换建立守恒量与Lagrange函数间的关系,从而求得系统的Lagrange函数,并讨论与守恒量相应的无限小变换的Noether对称性与Lie对称性,最后求得系统的运动学方程.     

CP symmetry and its violation in fundamental interactions

The concept of discrete symmetries in classical and quantum physics is reviewed. An account is given ofCP violation observed in theK-meson system and of other experiments whereCP symmetry has been tested. The present theoretical ideas onCP violation within the standard model, and problems needing extension of the model are described. Finally, ideas and experimental approaches toCP violation beyond the standard model are reviewed in brief.     

On conservation laws and boundary conditions for “short waves” equation

In this paper we study local conservation laws for the equation of short waves in the form of a variational problem. We analyze an infinite symmetry group of the equation and generate a finite number of conservation laws corresponding to given infinite symmetries through appropriate boundary conditions.     

The neutrino mass matrix and (selected) variants of <Emphasis Type="Italic">A</Emphasis><Subscript>4</Subscript>

Recent neutrino oscillation experiments have measured leptonic mixing angles with considerable precision. Many theoretical attempts to understand the peculiar mixing structure, observed in these measurements, are based on non-Abelian flavour symmetries. This talk concentrates exclusively on models based on the non-Abelian symmetry A ₄. A ₄ is particularly well suited to describe three family mixing, and allows to explain the near tri-bimaximal mixing observed. Special emphasis is put here on the discussion of the neutrinoless double beta decay observable 〈:m _ν 〉. Different models based on A ₄ with very similar predictions for neutrino angles can yield vastly different expectations for 〈m _ν 〉. Neutrinoless double beta decay can thus serve, in principle, as a discriminator between different neutrino mass models.      

Local symmetries in gauge theories in a finite-dimensional setting

It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group bundles and, more generally, of Lie groupoids and their actions on fiber bundles. This applies not only to local symmetries, which lie at the heart of gauge theories, but is already true even for global symmetries when one allows for fields that are sections of bundles with (possibly) non-trivial topology or, even when these are topologically trivial, in the absence of a preferred trivialization.     

Symmetry of Lyapunov spectrum

The symmetry of the spectrum of Lyapunov exponents provides a useful quantitative connection between properties of dynamical systems consisting ofN interacting particles coupled to a thermostat, and nonequilibrium statistical mechanics. We obtain here sufficient conditions for this symmetry and analyze the structure of 1/N corrections ignored in previous studies. The relation of the Lyapunov spectrum symmetry with some other symmetries of dynamical systems is discussed.     

Study on a General Hopf Hierarchy

By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.     

New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.     

The general solution of Bianchi type <Emphasis Type="Italic">V I I</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript> vacuum cosmology

The theory of symmetries of systems of coupled, ordinary differential equations (ODE) is used to develop a concise algorithm in order to obtain the entire space of solutions to vacuum Bianchi Einstein’s field equations (EFEs). The symmetries used are the well known automorphisms of the Lie algebra for the corresponding isometry group of each Bianchi Type, as well as the scaling and the time re-parametrization symmetry. The application of the method to Type V I I _h results in (a) obtaining the general solution of Type V I I ₀ with the aid of the third Painlevé transcendental P _{I I I} ; (b) obtaining the general solution of Type V I I _h with the aid of the sixth Painlevé transcendental P _{V I} ; (c) the recovery of all known solutions (six in total) without a prior assumption of any extra symmetry; (d) The discovery of a new solution (the line element given in closed form) with a G ₃ isometry group acting on T ₃, i.e., on time-like hyper-surfaces, along with the emergence of the line element describing the flat vacuum Type V I I ₀ Bianchi Cosmology.     

Spectral Degeneracies in the Totally Asymmetric Exclusion Process

We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at arbitrary filling. Although the system does not possess obvious symmetries except translation invariance, the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules. This behaviour is explained by a hidden symmetry property of the Bethe Ansatz. Assuming a one-to-one correspondence between the solutions of the Bethe equations and the eigenmodes of the Markov matrix, we derive combinatorial formulae for the orders of degeneracy and the number of multiplets. These results are confirmed by exact diagonalisations of small size systems. This unexpected structure of the TASEP spectrum suggests the existence of an underlying large invariance group.     

Symmetries and Similarity Reductions of a New （2＋1）-Dimensional Shallow Water Wave System

The symmetries of a （2＋1）-dimensional shallow water wave system, which is newly constructed through applying variation principle of analytic mechanics, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. The （1＋1） dimensional displacement shallow water wave equation can be derived from the reductions when special symmetry parameters are chosen.     

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.     

Symmetries of some classes of dynamical systems

In this paper a three-dimensional system with five parameters is considered. For some particular values of these parameters, one finds known dynamical systems. The purpose of this work is to study some symmetries of the considered system, such as Lie-point symmetries, conformal symmetries, master symmetries and variational symmetries. In order to present these symmetries we give constants of motion. Using Lie group theory, Hamiltonian and bi-Hamiltonian structures are given. Also, symplectic realizations of Hamiltonian structures are presented. We have generalized some known results and we have established other new results. Our unitary presentation allows the study of these classes of dynamical systems from other points of view, e.g. stability problems, existence of periodic orbits, homoclinic and heteroclinic orbits.