Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations

A straightforward algorithm for the symbolic computation of generalized (higher‐order) symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi‐discrete lattice equations. With our Integrability Package, generalized symmetries are obtained for several well‐known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Goldstone Boson Normal Coordinates in Interacting Bose Gases

Spontaneous symmetry breaking (SSB) is one of the basic aspects of collective phenomena such as phase transitions in statistical mechanics or ground-state excitations in field theory. In general, spectral analysis of SSB is related to the presence of a Goldstone boson particle. The explicit construction of the canonical variables (boson field operator and its adjoint) of this boson has so far been an open problem. In this paper, we consider the SSB of Bose–Einstein condensation in two models: the so-called imperfect or mean field Bose gas (which is nothing but a perfect ideal Bose gas including the property of equivalence of ensembles), and the Bogoliubov weakly interacting Bose gas. For both we construct explicitly the Goldstone boson field variables. The remarkable result is that in both cases the field and its adjoint field are formed as the fluctuation operators respectively of the order parameter operator and of the generator of the broken symmetry. The notion of fluctuation operator is essential for our mathematical construction. We find that although the order parameter has a critical fluctuation, the generator of the broken symmetry has a squeezed fluctuation of the same inverse rate. Furthermore, we prove that this canonical pair of variables decouples from the other variables of the system, and that these fluctuations behave dynamically as long-wavelength sound waves or as oscillator variables.     

对称性和热学

试图探索一条不同于传统做法的、概括和表述热学基本定律的途径,从理论上,把热学置于对称性原理的基础之上,加以概括和解释。     

Paramagnetic Proton Nuclear Spin Relaxation Theory of Low-Symmetry Complexes for Electron Spin Quantum NumberS=

A generalization of the modified Solomon–Bloembergen–Morgan (MSBM) equations has been derived in order to describe paramagnetic relaxation enhancement (PRE) of paramagnetic complexes characterized by both a transient (Δ^ZFS_t) and a static (Δ^ZFS_s) zero-field splitting (ZFS) interaction. The new theory includes the effects of static ZFS, hyperfine coupling, and angular dependence and is presented for the case of electron spin quantum numberS= , for example, Mn(II) and Fe(III) complexes. The model gives the difference from MSBM theory in terms of a correction term δ which is given in closed analytical form. The theory may be important in analyzing the PRE of proton spin–lattice relaxation dispersion measurements (NMRD profiles) of low-symmetry aqua–metal complexes which are likely to be formed upon transition metal ions associated with charged molecular surfaces of biomacromolecules. The theory has been implemented with a computer program which calculates solvent water protonT₁NMRD profiles using both MSBM and the new theory.     

Coupling Parameters and the Form of the Potential via Noether Symmetry

We explore the conditions for the existence of Noether symmetries in the dynamics of FRW metric, non minimally coupled with a scalar field, in the most general situation, and with nonzero spatial curvature. When such symmetries are present we find a general exact solution for the Einstein equations. We also show that non Noether symmetries can be found. Finally, we present an extension of the procedure to the Kantowski-Sachs metric which is particularly interesting in the case of degenerate Lagrangian.     

The Kramers–;Wannier Symmetry and S-Duality in the Two-Dimensional g03A6;4 Theory

We show that the exact beta function of the two-dimensional g⁴ theory possesses two dual symmetries. These are the Kramers–Wannier symmetry d(g) and the strong–weak-coupling symmetry, or the S-duality f(g), connecting the strong- and weak-coupling domains lying above and below the fixed point g ^*. We obtain explicit representations for the functions d(g) and f(g). The S-duality transformation f(g) allows using the high-temperature expansions to approximate the contributions of the higher-order Feynman diagrams. From the mathematical standpoint, the proposed scheme is highly unstable. Nevertheless, the approximate values of the renormalized coupling constant g ^* obtained from the duality equations agree well with the available numerical results.     

High-Energy Relations Between Elastic Scattering Amplitudes of Particles and Antiparticles

We find conditions under which the ratio of particle and antiparticle total scattering cross sections as s. If the forward elastic scattering amplitudes become purely real asymptotically, then their ratio tends to –1. We prove that /_–(s)1 for ₊(s)/m ², where m is the -meson mass. We show that the asymptotic relations obtained have finite-energy analogues for some processes.     

Some Non-trivial Symmetry Classes of Tensors Associated with Certain Characters

Let G be a finite group and a set of n elements. Assume that G acts faithfully on and let V be a vector space over the complex field , with dim V = m 2. It is shown that for each irreducible constituent of permutation character of G, the symmetry class of tensors associated with G and is non-trivial. This extends a result of Merris and Rashid (see [6, Theorem 2]).1995 AMS subject classification primary 20C30 secondary 15A69This research was in part supported by a grant from IPM.     

The replica-symmetric solution without replica trick for the Hopfield model

We derive the saddle-point equations for the order parameters of the Hopfield model in the case of replica symmetry without using the replica trick, but assuming that the Edwards-Anderson parameter is a self-averaging quantity.