Some comments on nonlinear dynamics

I summarize here the remarks made at the closing of the Conference and Research Workshop: Perspectives on Nonlinear Dynamics, held in Trieste in July 2007.      

Some comments on perturbation theory

It is shown how, with no significant extra effort, the results of a third-order perturbation calculation may be used to calculate a least upper bound to the energy. Examples suggest that this bound is a better approximation than the usual perturbation series to third order. When only second-order calculations are made it is shown how a modified formula gives a much better approximation than the usual series to second order, again with no significant extra effort.     

Some comments on quasi-bound-state calculations

Quasi-bound-state energy calculations are made for a perturbed oscillator and a perturbed hydrogen atom. The theory of the least-squares method is analysed. A simple numerical variant of the well-known stabilization method is tested, and the results are compared with those from the least-squares and perturbation-series methods.     

关于非球面制造技术的看法

列举了现代诸高科技领域对非球面光学系统的需求实例，如短波光学，空间光学，红外光学，微电子技术等。然后比较讨论了刀口阴影法与数字干涉检验的优缺点。在非球面修磨方法方面，首先肯定了计算机辅助加工是发展的方向，指出其主要优点在于“失误”造成的反复率大大降低。最后对非球面技术提出了总的看法，并对检测方法和修磨方法提出了几个研究课题。     

Some comments on band contour methods

The validity of the band contour method for obtaining rotational constants from unresolved bands is examined, particularly with regards to the matching of intensities for the contour. This question is reviewed in the light of current theories of band models, and suggestions and recommendations are made for valid procedures for intensity comparisons. A brief summary of available programs for the theoretical calculation of contours is given.     

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291     

Some comments on the many-dimensional solitons

The possibilities for the existence of truly localized soliton solutions in the realistic three-dimensional case are discussed. The gauge invariant theory of a non-linear chiral field is shown to be a good candidate for a model with solitons.     

Some comments on the photomagnetoelectric effect

A generalization of the theory for the photomagnetoelectric effect (PME) is reported. Experimental quantities such as short circuit current, open circuit voltage and magnetoresistance are described as functions of carrier lifetime, surface velocities at the sample surfaces, electron and hole mobilities, electron and hole concentrations, absorption and quantum efficiency coefficients, intensity of light, magnetic field and sample dimensions. The results of numerical computation are also presented.     

Some comments on bosonisation and biproducts

We collect here some less well-known results and formulae about the bosonisation construction which turns braided groups into quantum groups. We clarify the relation with biproduct Hopf algebras (the constructions are not the same), the response to twisting of braided groups and the abstract characterisation via automorphisms of the forgetful functor for the category of (co)modules of a braided group.     

Some comments on the non-self-dual Nahm equations

Some empirical results on the cubic algebra are presented. The algebra is satisfied at the residue of any pole in a solution to Nahm's non-self-dual equations.Sir Derman Christopherson Foundation Fellow     

Some comments on constrained control of chaos

Only two conditions were considered in paper titled as Constrained control of chaos appeared in [Y.-C. Tian, M.O. Tade, D. Levy, Phys. Lett. A 296 (2002) 87], in this Letter the third condition was toughed upon by their ways, and a special problem which was unlike the already results was investigated and the corresponding solving strategy was put forward, simulation results were also provided to verify the suggested scheme.     

对高平均功率DPL的几点认识

高平均功率二极管泵浦激光(DPL)的发展历程充满了技术创新,解决热管理问题的不同思路导致了各种不同的具体技术路线,其中包含着空间域、时间域和材料学的创新,甚至超越了全固态激光的范畴.指出了高平均功率DPL有广阔的应用前景,军民两用,类型多样.影响高平均功率DPL应用的制约因素主要是能源、有效热控等问题.     

Some comments on the Coriolis attenuation problem

To explore the Coriolis attenuation problem we have carried out a schematici _13/2 rotor plus single quasi-particle band-mixing calculation. The results reveal that the calculations are largely insensitive towards the location of the Fermi energy near the low-K single particle states only, and therefore are incapable of taking into account the transition from ‘full’ decoupling to ‘partial’ decoupling as the Fermi level is increased. We trace the possible reasons for this insensitivity and find that this may be primarily due to thebcs approximation for calculating the quasiparticle energies.     

Some comments on modeling the linearized Boltzmann equation

Some exact solutions of the homogeneous and the inhomogeneous linearized Boltzmann equation (LBE) for rigid-sphere collisions are used to define two model equations in the general area of rarefied-gas dynamics. These equations are obtained from a systematic development of two synthetic scattering kernels that yield model equations that have as exact solutions certain known exact solutions of the homogeneous and of the inhomogeneous LBE. The first model established is defined in terms of the collisional invariants and the Chapman-Enskog integral equations for viscosity and for heat conduction. An extended model is defined also in terms of the collisional invariants and the Chapman-Enskog functions for viscosity and heat conduction, but the first and second Burnett functions are also included in the model. The variable collision frequency or generalized BGK model is also obtained as a special case. In addition, the exact mean-free paths defined, for rigid-sphere collisions and the LBE, in terms of viscosity or heat conduction are employed to define approximations of these quantities that are consistent with the use of the variable collision frequency model.     

Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex ^N y ^N+x ^N+y ^N+1/k ²=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.     

Some comments on approximations to the master equation

A common strategy for approximating a master equation is to replace it by a diffusion-like equation. Many methods for deriving the form of such an equation have been suggested in the literature. We compare two of these in the light of an example in which the master equation can be solved exactly. One of these is the van KampenΩ-expansion, which generally does not give a useful approximation to the equilibrium solution, and the second is a technique which preserves the noise-free dynamics and gives the correct equilibrium solution. It is shown that the second moment calculated in the latter approximation is not an accurate one at short times. The difficulty is the restriction of the approximating equation to the diffusion form.