Chaos in PDEs and Lax Pairs of Euler Equations

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1) transversal homoclinic orbits, (2) Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite-dimensional autonomous systems, the author was able to prove the existence of chaos through a shadowing lemma. Around the Silnikov homoclinic orbits, the author was able to prove the existence of chaos through a horseshoe construction.Very recently, there has been a breakthrough by the author in finding Lax pairs for Euler equations of incompressible inviscid fluids. Further results have been obtained by the author and collaborators.     

Frobenius Pairs with Perfect Involutions

An involution i of a group G is said to be perfect in G if any two non-commuting involutions in i^G are conjugated by an involution in the same class. We generalize theorems of Jordan and M. Hall concerning sharply doubly transitive groups, and the Shunkov theorem on periodic groups with a finite isolated subgroup of even order. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 751–762, November–December, 2005.     

Regularities of triplet-triplet annihilation of aromatic hydrocarbons in aqueous micellar solutions of sodium dodecyl sulfate

Investigation of deactivation processes in triplet states of a series of aromatic hydrocarbon molecules (antracene, 1,2-benzantracene, and 3,4-benzpyrene) made it possible to reveal the presence of triplet-triplet annihilation of the molecules in aqueous micellar solutions of sodium dodecyl sulfate. It is shown that the effect of microheterogeneous solutions on the process of triplet-triplet annihilation manifests itself in an increase in the probability of excimerization upon dissociation of triplet pairs of 1,2-benzantracene and 3,4-benpyrene molecules compared to one-component solutions. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 64, No. 5, pp. 626–628, September–October, 1997.     

Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators—the case of the compact type

A new notion which is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair can be defined by generating functions. In the case of the compact type, this statement is shown. This notion also concerns an existence of a “good” parametrization for the spherical representations. These mean that generating functions can be dealt with systematically in the compact case.     

Simultaneous determination of preservatives,sweeteners and antioxidants in foods by paired-ion liquid chromatography

Summary A paired-ion, reversed-phase high-performance liquid chromatographic method was developed for the simultaneous determination of sweeteners (dulcin, saccharin-Na and acesulfame-K), preservatives (sodium dehydroacetate, sorbic acid, salicyclic acid, benzoic acid, succinic acid, methyl-para-hydroxybenzoic acid, ethyl-para-hydroxybenzoic acid, n-propyl-para-hydroxybenzoic acid, isopropyl-para-hydroxybenzoic acid, n-butyl-para-hydroxybenzoic acid, and isobutyl-para-hydroxybenzoic acid), and antioxidants (3-tertiary-butyl-4-hydroxyanisole and tertiary-butyl-hydroquinone). A mobile phase of acetonitrile-50 mM aqueous α-hydroxyisobutyric acid solution (pH 4.5) (2.2 : 3.4 or 2.4 : 3.6, v/v) containing 2.5 mM hexadecyltrimethylammonium bromide and a C₁₈ column with a flow rate at 1.0 mL/min and detection at 233 nm were used. This method was found to be very reproducible with detection limits ranged from 0.15 to 3.00 μg. The retention factor (k) of each additive could be affected by concentrations of hexadecyltrimethylammonium bromide and α-hydroxyisobutyric acid, and pH and ratio of mobile phase. The presence of additives in some food samples was determined.     

Geng族可积发展方程的Lax表示

基于曹的思想,我们提供—族由耿献国(Geng Xianguo)研究的可积发展方程的Lax表示。     

Dirac定理的局部化与Hamilton图

设Ｇ为一个ｎ阶２－连通图，ｎ≥３．若｜Ｄ_ｎ／２（Ｋ_１，３）｜≥２且满足下述条件之一：ｉ）｜Ｄ_ｎ／２（Ｋ_１，３＋ｅ）｜≥２，ii）若Ｋ_１，３＋ｅ→Ｇ，ｘｙ(?)Ｅ（Ｋ_１，３＋ｅ），则ｍａｘ｛ｄＧ（ｘ），ｄＧ（ｙ）｝≥ｎ／２，则Ｇ是一个Ｈａｍｉｌｔｏｎｉａｎ图或其闭包为ｓＰ|⊕Ｈ，这里ｓＰ⊕Ｈ是一类极小２－边连通图．     

Compatible Poisson Brackets on Lie Algebras

We discuss the relationship between the representation of an integrable system as an L-A-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.     

RESTRICTED FLOWS OF A HIERARCHYOF INTEGRABLE DISCRETE SYSTEMS

1.IntroductionTherestrictedflowsofsolitonhierarchyhavebeenextensivelystudied(see,forexample,[1--7]).Theapproachforconstructingrestrictedflowsofsolitonhierarchycanalsobeappliedtoobtainrestrictedflows(discretemaps)ofahierarchyofdiscreteintegrablesystems(nonlineardifferential-differenceequations)IS,9].TheserestrictedflowshavetheformofLagrangeequationsandthereforecanmodelphysicallyinterestingprocesses.Wesupposethatthehierarchyofdiscreteintegrablesystems(DIS)isassociatedwithadiscreteisospectralp…     

A study of pion absorption by16O using realistic interactions—I. Twonucleon emission following bound-pion absorption

Matrix element of the Galilean invariant non-relativistic reduction of the pseudoscalar-pseudovector interaction has been calculated assuming the reaction to be a direct process with boundπ ⁻ being absorbed by a correlated pair of nucleons. The Hartree-Fock wavefunctions obtained with the unitary-model-operator approach starting with the realistic nucleon-nucleon interaction have been used forπ-capturing nucleon pair in the initial state. The calculations have been done with and without antisymmetrising the initial state wavefunction of the pion absorbing pair. For the final state nucleon-nucleon interaction has been taken into account. The strongπ-nucleus interaction together with the Coulomb interaction with the finite nuclear size on the bound pion wavefunction are taken into account. Angular distributions of the emitted nucleon-pair, the branching ratios and the total absorption rates are calculated for¹⁶O with and without antisymmetrisation effect. The calculated results are compared with the experimental and other theoretical work.