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.

     

Prediction of the retention indices of methyl pyridines and pyrazines in capillary gas chromatography based on the non-linear additivity of the sorption energy

Summary Methylbenzenes, pyridines and pyrazines were investigated on fused-silica and glass capillary columns coated with SE-30 and PEG-40M/KF liquid phases, at two temperatures, 80° and 110°C. The contribution of the methylene groups to the partial molar free sorption energy was determined for methylpyridines and pyrazines. Equations are proposed for the calculation of the retention indices of methyl pyridines and pyrazines. These equations are based on the ortho- and α-effects of the methyl groups. The predicted indices have been experimentally tested for six dimethyl- and trimethylpyridines, and four methylpyrazines. Good accuracy of the calculation permits to use this method for the identification of methylpyridines and pyrazines in complex mixtures. Enlarged text of the paper presented at the Eighth International Symposium on Capillary Chromatography, Riva del Garda, Italy, May 19–21, 1987.     

Effects of solute-matrix interaction on monitoring the conformational changes of immobilized proteins by surface plasmon resonance sensor

A real-time and labeling-free surface plasmon resonance (SPR) sensor was used to monitor the conformational changes of immobilized globule proteins (RNase A and lysozyme) in chemical unfolding and refolding. The effects of chemical denaturants on the protein structures were investigated. The methodology in protein conformational study on the solid surface is refined through the theoretic calculations and the conformational information of native/denatured proteins in solution. Additionally, our observation illustrates that the ambient buffer solution is merit to influence the refractive index of immobilized protein films and directly be observed from the SPR resonance angle shifts.     

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.     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

A refined physico-chemical model of solute retention in reversed-phase high-performance liquid chromatography with ternary mobile phases

Summary In our previous publication we have introduced a new model of solute retention in RP-HPLC systems with ternary mobile phases of the B+AB₁+AB₂ type (B: acetonitrile or tetrahydrofuran; AB₁: methanol; AB₂: water). That model proposed no stoichiometric differentiation between acetonitrile and tetrahydrofuran, alternatively present in the solvent system; moreover, it made some very rough assumptions only as to the intermolecular interactions among the mobile phase constituents.This paper introduces a significant refinement to the already established retention model, which is based on the simple quantitative relationships between acetonitrile and tetrahydrofuran, and the remaining components of the ternary liquid system. The refined model is tested with same experimental data.     

双柱气相色谱法测定有机物的正辛醇/水分配系数

通过对烷烃，醇类等有机物在五种不同极性固定相上的气相色谱保留值的测定，提出了从有机物气相色谱保留值测定其正辛醇/分配系数的公式，研究了从不同极性固定相上测得的保留值来测定其分配系数的准确性，得出烷烃类在所有柱上测得的保留值与其分配系数间均具有很好的关联，烷烃醇类在极性相异柱上的保留值之比值与其分配系数间也有很好的相关性，从而为烷烃及醇类等有机物分配系数的测定提供了一种简单、可靠，易行的新方法。     

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.     

Light induced refractive index changes in PMMA films doped with styrene

Solutions of poly(methyl methacrylate) in styrene are used to produce thin polymer films (1m–20m) saturated with styrene monomer. In the illuminated areas refractive index changes up to 10^–2 are achieved by photoinduced polymerisation. The index patterns may be fixed by annealing treatments removing the residual monomer.We thank P. Hertel, M. Kopietz, M. D. Lechner and D. Steinmeier for valuable discussions. Financial support of the Stiftung Volkswagenwerk is gratefully acknowledged.