From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.     

Measurement of Large Dipolar Couplings of a Liquid Crystal with Terminal Phenyl Rings and Estimation of the Order Parameters

NMR spectroscopy is a powerful means of studying liquid‐crystalline systems at atomic resolutions. Of the many parameters that can provide information on the dynamics and order of the systems, ¹H–¹³C dipolar couplings are an important means of obtaining such information. Depending on the details of the molecular structure and the magnitude of the order parameters, the dipolar couplings can vary over a wide range of values. Thus the method employed to estimate the dipolar couplings should be capable of estimating both large and small dipolar couplings at the same time. For this purpose, we consider here a two‐dimensional NMR experiment that works similar to the insensitive nuclei enhanced by polarization transfer (INEPT) experiment in solution. With the incorporation of a modification proposed earlier for experiments with low radio frequency power, the scheme is observed to enable a wide range of dipolar couplings to be estimated at the same time. We utilized this approach to obtain dipolar couplings in a liquid crystal with phenyl rings attached to either end of the molecule, and estimated its local order parameters.     

A 1H NMR and theoretical investigation of the conformations of some monosubstituted cyclobutanes

The complete analysis of the complex (1)H NMR spectra of some monosubstituted cyclobutanes was achieved to give all the (1)H chemical shifts and (n)J(HH) (n = 2, 3 and 4) coupling constants in these molecules. The substituent chemical shifts of the substituents in the cyclobutane ring differ significantly from those in acyclic systems. For example, the OH and the NH(2) groups in cyclobutanol and cyclobutylamine produce a large shielding of the hydrogens of the opposite CH(2) group of the ring compared with little effect on the comparable methylene protons of butane. These effects and the other (1)H shifts in the cyclobutanes were modelled successfully in the CHARGE program. The RMS error (calculated vs observed shifts) for the 34 (1)H shifts recorded was 0.053 ppm. The conformational equilibrium in these compounds between the axial and the equatorial conformers was obtained by comparing the observed and the calculated (4)J(HH) couplings. These couplings in cyclobutanes, in contrast to the corresponding (3)J(HH) couplings, show a pronounced orientation dependence; (4)J(eq-eq) is ca 5 Hz and (4)J(ax-ax) ca 0 Hz. The couplings in the individual conformers were calculated at the B3LYP/EPR-III level. The conformer energy differences ΔG(ax-eq) vary from 1.1 kcal mol(-1) for OH to 0.2 kcal mol(-1) for the CH(2)OH substituent. The values of the conformer energy differences are compared with the previous IR data and the corresponding theoretical values from molecular mechanics (MM) and DFT theory. Generally, good agreement is observed although both the MM and the DFT calculations deviate significantly from the observed values for some substituents.     

Darboux transformation and explicit solutions for the integrable sixth-order KdV equation for nonlinear waves

Korteweg-de Vries (KdV)-type equations can describe some physical phenomena in fluids, nonlinear optics, quantum mechanics, plasmas, etc. In this paper, with the aid of symbolic computation, the integrable sixth-order KdV equation is investigated. Darboux transformation (DT) with an arbitrary parameter is presented. Explicit solutions are derived with the DT. Relevant properties are graphically illustrated, which might be helpful to understand some physical processes in fluids, plasmas, optics and quantum mechanics.     

Stabilization of novel multi-layer networks with noise-based nonlinear superior couplings via aperiodically adaptive intermittent pinning control

In this paper, novel multi-layer networks with superior couplings are proposed firstly which are established on a non-strongly connected digraph. Within the multi-layer networks, a nonlinear coupling based on white noises is introduced, which is the feature of superior couplings. We adopt aperiodically adaptive intermittent pinning control to stabilize the multi-layer networks. An concrete analysis framework about selecting the target vertex of the control is revealed. Aperiodically adaptive intermittent control is employed on the vertex systems of the first layer networks, to achieve the stabilization of the first layer networks, where the couplings of drift terms are treated as negative effects on stabilization. With the help of noise stabilization, the stabilization of the other layers networks is realized based on the stability of the first layer networks and the characteristics of the superior coupling that is based on white noises. By employing graph theory and the Lyapunov method, an almost sure exponential stabilization criterion of the multi-layer networks is acquired. As a subsequent result, the proposed theory is applied to a class of stochastic coupled oscillators with sufficient conditions being given to ensure their stability. Finally, a numerical example is provided to illustrate the feasibility of the stated theoretical results.     

Intermolecular multiple-quantum coherence NMR signals modulated by double distant dipolar fields

A modified CRAZED pulse sequence was applied to obtain the intermolecular multiple-quantum coherence NMR signals from double distant dipolar fields in highly polarized spin systems. Complete theoretical analyses were explicitly derived from the dipolar field treatment combined with product operator formalism. Two typical samples containing several different components were chosen for the experimental verifications. The computer simulations and experimental observations are consistent with the theoretical predictions. The results presented herein provide a convenient way to understand the combined effects of multiple distant dipolar fields from the different components in complicated chemical or biological solutions. When experimental conditions such as selective radio-frequency pulses are not optimal, it may be helpful to identify possible unexpected signals or artefacts of high-resolution NMR spectroscopy in inhomogeneous fields.     

密度矩阵重正化群的异构并行优化

密度矩阵重正化群方法（DMRG）在求解一维强关联格点模型的基态时可以获得较高的精度,在应用于二维或准二维问题时,要达到类似的精度通常需要较大的计算量与存储空间.本文提出一种新的DMRG异构并行策略,可以同时发挥计算机中央处理器（CPU）和图形处理器（GPU）的计算性能.针对最耗时的哈密顿量对角化部分,实现了数据的分布式存储,并且给出了CPU和GPU之间的负载平衡策略.以费米Hubbard模型为例,测试了异构并行程序在不同DMRG保留状态数下的运行表现,并给出了相应的性能基准.应用于4腿梯子时,观测到了高温超导中常见的电荷密度条纹,此时保留状态数达到10⁴,使用的GPU显存小于12 GB.     

Central elements of the elliptic monodromy matrix algebra at roots of unity

The central elements of the algebra of monodromy matrices associated with the R-matrix are studied. When the crossing parameter w takes a special rational value , where N and n are positive coprime integers, the center is substantially larger than that in the generic case for which the “quantum determinant” provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo.