The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

A tight upper bound for the number of intersections between two rectangular paths

The problem of finding the number of intersections between two geometric figures in the plane has been studied extensively in literature. In this paper, the geometric figure comprising a continuous rectilinear path (called rectangular path) is considered, and a tight (least) upper bound onI(P, Q), the number of intersections between two rectangular pathsP andQ, is given.Editors' Note: One of our referees has reported that the main result of this paper has recently been given independently by a Chinese researcher at the University of Science and Technology, Hefei, P. R. of China. His paper is under publication in the Chinese Science Bulletin. However, since this journal may not be easily accessible to our readers, and further the two papers are obviously independent of each other, theBIT Editors have decided to accept the present paper.     

ASYMPTOTIC APPROXIMATION METHOD AND ITS CONVERGENCE ON SEMI-INFINITE PROGRAMMING

The aim of this article is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented by making use of two general discrete approximation methods. Simultaneously, the consistence and the epi-convergence of the asymptotic approximation problem are discussed.     

Packing two graphs of ordern having total size at most 2n − 2

Two graphsG andH of the same order are packable ifG can be embedded in the complement ofH. In this paper we give a complete characterization of two graphs of ordern having total size at most 2n – 2 which are packable. This result extends an earlier result of B. Bollobás and S.E. Eldridge.     

Crystallinity of ethylene oxide oligomers

Nonaethylene glycol and pentadecaethylene glycol and their dimethyl ethers have been prepared and characterized, with respect to crystallinity by wide- and small-angle X-ray scattering, Raman scattering, i.r. spectroscopy and differential scanning calorimetry. Wide-angle X-ray scattering is similar to that from high molecular weight poly(ethylene oxide). The crystal habit is lamella. The lamellae are highly crystalline and the surface layers are ordered. Comparison with crystalline poly(ethylene oxide) prepared conventionally, and having a distribution of chain lengths, shows that such samples crystallize into lamellae with disordered surface layers.     

Synthesis and Structure of [DB24C8)Na][Cd(SCN)3. Formation of a novel linear Cd...Cd...Cd chain with a mer-CdN3S3 coordination confirmation and new coiled [(DB24C8)Na]+ cation

We report herein a novel coordination solid, [(DB24C8)Na][Cd(SCN)3] (6) (DB24C8 denotes dibenzo-24-crown-8), which exhibits a new type of [Cd(SCN)3-]infinity chain with two unusual stereochemical characteristics: (1) a mer-CdN3S3 coordination and (2) a linear Cd chain with a Cd...Cd...Cd angle of 180 degrees. In addition, the [(DB24C8)Na]+ monocation adopts a new structural type-a coiled structure-for the combination of crown ether DB24C8 and alkali metal Na+. The title compound crystallizes in a monoclinic unit cell of C2/c space group symmetry with lattice parameters a = 16.110(8) A, b = 20.380(5) A, c = 11.01(1) A, beta = 119.87(3) degrees, and Z = 4. The arrangement of the [Cd(SCN)3-](infinity) chains in the crystal lattice in the title compound is approximately hexagonal, creating triangular channels which are filled with [(DB24C8)Na]+ monocations. It was previously reasoned by us that the coiled [(DB24C8)Na]+ monocation, which lacks inversion or mirror symmetries, should enhance the tendency for the formation of the noncentrosymmetric space group of the title crystal, making it a potential second-order nonlinear optical crystal. Interestingly, however, the title compound crystallizes in a centrosymmetric space group (C2/c) and gives rise to no second harmonic generation (SHG). Previously known [Cd(SCN)3-](infinity) chains adopt fac-CdN3S3 coordination and a zigzag Cd chain configuration with a Cd...Cd...Cd angle of 165 degrees. The zigzag chains can align in either parallel or antiparallel fashion, resulting in efficient or no SHG effects, respectively. The linear Cd.Cd.Cd chain configuration observed in the title compound, on the other hand, makes it indistinguishable between parallel and antiparallel alignments. It is concluded that, to ensure the formation of noncentrosymmetric space groups, it is necessary to employ optically pure chiral cations as spacers and/or controllers. Furthermore, to enhance the nonlinear optical responses, [Cd(SCN)3-]infinity chains with fac-CdN3S3 coordination and parallel alignments of the zigzag Cd chains should be used.     

Preparation of a piezoelectric immunosensor for the detection of Salmonella paratyphi A by immobilization of antibodies on electropolymerized films

 To obtain new coatings for the preparation of piezoelectric immunosensors, the anodic polymerization of o-aminophenol (oAP), o-phenylenediamine (oPD) and m-phenylenediamine (mPD) onto a gold-plated crystal has been studied. The possibility of immobilizing an antibody (anti-S. paratyphi A) onto the crystal via the electropolymerized films was investigated. The poly-mPD film gave the best results for immobilizing the antibody. With antibodies crosslinked on the poly-mPD film, a piezoelectric immuno-assay for the detection of S. paratyphi A was proposed. The shift (ΔF=F_20s−F_400s, Hz) between the frequency at 20 s after the addition of sample (F_20s), and that of 400 s (F_400s) was used to construct a calibration graph, and shortening of the assay time was achieved. The S. paratyphi A concentration in the range of 10^5–10⁹ cells/ml can be measured by this method. Received: 9 May 1996/Revised: 6 September 1996/Accepted: 19 September 1996     

胆红素在纳米TiO2膜上的吸附行为研究

采用石英晶体微天平现场技术,研究了胆红素在纳米TiO2表面的吸附动力学行为,并考察了溶液pH、硅烷亲脂性试剂及金属离子修饰的纳米TiO2对胆红素吸附的影响。结果表明,胆红素在纳米TiO2表面的吸附平衡常数K为2·0×106L/mol;由于亲水性纳米TiO2表面的羟基和胆红素分子中的羧基发生作用,胆红素吸附量随溶液pH增大而增大;但经过硅烷试剂亲脂性修饰后,吸附量随pH增大而下降;经过金属离子修饰后,改变了TiO2表面电荷导致吸附量增加,且吸附量随金属离子电荷增加而增加,当金属离子所带电荷相同时,吸附量基本相同。     

Catalytic enantioselective allylation of aldehydes via a moisture-tolerant chiral BINOL-In(III) complex

A moisture-tolerant chiral indium complex has been developed to effect good enantioselectivities in the addition of allyltributylstannanes to aldehydes. The allylation of a variety of aromatic, α,β-unsaturated and aliphatic aldehydes resulted in both moderate to good yields and high enantioselectivities (up to 86% ee).