Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

Geometric indices and the Alexander polynomial of a knot

It is well-known that any Laurent polynomial satisfying and is the Alexander polynomial of a knot in . We show that can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

     

Correlations between Van der Waals' volume and retention index general equation applicable to different homologous series

Summary Correlations between retention index (R_I) and Van der Waals' volume (V_W) in homologous series of n-alcohols, n-aldehydes and esters are studied on the stationary phases Carbowax 1540 and squalane. Influence of functional groups is quantified and a method for obtaining one general equation, valid for an indeterminate number of homologous series, is proposed.     

Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes

Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita’s proligand method formulated in Parts 1–3 of this series [Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously [Fujita in Tetrahedron 47:31–46, 1991], a planted promolecule is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands. Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating functions are based on cycle indices with chirality fittingness (CI-CFs), which are composed of three kinds of sphericity indices (SIs), i.e., a _d for homospheric cycles, c _d for enantiospheric cycles, and b _d for hemispheric cycles. For the purpose of evaluating c _d recursively, the concept of diploid is proposed, where the nested nature of c _d is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x ²), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem initiated by mathematician Cayley [Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

烯烃聚合数学模拟研究与现状

综述了近年来国内外在烯烃聚合微观形态和宏观反应器数学模拟方面取得的研究成果，指出了烯烃聚合过程数学模拟的发展方向。     

Ab initio Valence Bond Study on AB-type Molecules,A Description for XH(X=L,Be,B,C,N,O,F) and XF(X=Li,Be,B)

Ab initio valence bond method is employed to quantitatively study the concepts of ionic resonance energy and ionicity of a chemical bond in the cases of hydrides XH(X=Li,Be,B,C,N,O,F) and fluorides XF(X=Li,Be,B),By establishing the relationship between resonance and stability,and comparing the calculated ionicities with Pauling‘s earlier estimations in the above diatomic molecules,the merits of Pauling‘s classical resonance theory were demonstrated at the ab initio level.     

烷基极化效应与X=O键伸缩振动频率

烷基取代物R’X=0的X=0键伸缩振动频率ν与烷基R的极化效应指数PEI(R)的关系可表示为:ν=a bPEI(R)。研究结果表明,烷基的极化效应使X=0键的伸缩振动频率降低。     

烯烃加成反应活性与烷基极化效应的关系

烯烃的自由基加成和亲电加成速率受不同的因素控制.对于仅仅含烷基取代的简单烯烃:(1)与OH自由基加成速率-log kOH可由π键上烷基极化效应指数的几何平均值GMPEIπ关联:-log kOH=a bGMPEIπ c(GMPEIπ)2; (2)与亲电试剂加成速率log k可由π键两端碳原子所连基团极化效应指数PEI(L)和PEI(S)关联:log k=a bPEI(L) cPEI(S),其中的PEI(L)和PEI(S)分别表示极化效应较大和较小的一端.     

离子性指数、极化效应指数烷烃^13C NMR化学位移的 关系研究

定义了烷烃分子中碳原子的离子性指数(INI)，用离子性指数(INI)、极化效应指数(PEI)及N^i~H(i=αβΥ)结构信息参数研究了烷烃的^13CNMR化学位移模型，结果表明，烷烃^13CNMR化学位移(CS)可用下式来定量描述：CS=194.6156-37.7394(INI)+98.6505(ΣPEI)+27.1630(INI/ΣPEI)-652.9106(ΣPEI/INI)+0.7735N^α~H+2.2468N^β~H-0.1742N^γ~H。用上式估算了304个碳原子的化学位移，平均绝对误差仅为0.77δ，标准差0.9860δ，预测值与实验值非常吻合。     

Densities,Ultrasonic Speeds,Viscosities and Refractive Indices of Binary Mixtures of Benzene with Benzyl Alcohol,Benzonitrile, Benzoyl Chloride and Chlorobenzene at 303.15 K

Densities,ρ, ultrasonic speeds, u, viscosities,η, and refractive indices, n, of pure benzene, benzyl alcohol (BA), benzonitrile (BN), benzoyl chloride (BC), chlorobenzene (CB) and their thirty six binary mixtures, with benzene as common component, were measured at 303.15 K over the entire mole fraction range. From these experimental data the values of deviations in ultrasonic speed, △u, isentropic compressibility, △k_s,excess acoustic impedance, Z^E, deviation in viscosity, Dh, and excess Gibbs free energy of activation of viscous flow, G^*E, and partial molar isentropic compressibility, Kφ,2⁰ of BA, BN, BC and CB in benzene were computed. The variation of these derived functions with composition of the mixtures suggested the increased cohesion (molecular order) in the solution and that interaction (A-B)＞(A-A) or (B-B).Moreover, theoretical prediction of ultrasonic speed, viscosity and refractive index of all the four binary mixtures was made on the basis of empirical and semi-empirical relations by using the experimental values of the pure components. Comparison of theoretical results with the experimental values was made in order to assess the suitability of these relations in reproducing the experimental values of u, η and n. Also, molecular radii of pure liquids and the average molecular radii of binary mixtures were evaluated using the corresponding refractive indices of pure liquids and binary mixtures. The average molecular radii of binary mixtures were found to be additive with respect to mole fraction of the pure component.