Generalized Rotation numbers

A rooted graph is a pair (G,x), where G is a simple undirected graph and x ∈ V(G). If G is rooted at x, its kth rotation number h_k (G,x) is the minimum number of edges in a graph F of order |G| + k such that for every v ∈ V(F) we can find a copy of G in F with the root vertex x at v. When k = 0, this definition reduces to that of the rotation number h(G,x), which was introduced in [“On Rotation Numbers for Complete Bipartite Graphs,” University of Victoria, Department of Mathematics Report No. DM-186-IR (1979)] by E.J. Cockayne and P.J. Lorimer and subsequently calculated for complete multipartite graphs. In this paper, we estimate the kth rotation number for complete bipartite graphs G with root x in the larger vertex class, thereby generalizing results of B. Bollobás and E.J. Cockayne [“More Rotation Numbers for Complete Bipartite Graphs,” Journal of Graph Theory, Vol. 6 (1982), pp. 403–411], J. Haviland [“Cliques and Independent Sets,” Ph. D. thesis, University of Cambridge (1989)], and J. Haviland and A. Thomason [“Rotation Numbers for Complete Bipartite Graphs,” Journal of Graph Theory, Vol. 16 (1992), pp. 61–71]. © 1993 John Wiley & Sons, Inc.     

Particle production from signature change

We consider the (massless) scalar field on 2-dimensional manifolds whose metric changes signature and which admit a spacelike isometry. Choosing the wave equation so that there will be a conserved Klein-Gordon product implicitly determines the junction conditions one needs to impose in order to obtain global solutions. The resulting mix of positive and negative frequencies produced by the presence of Euclidean regions depends only on the total width of the regions, and not on the detailed form of the metric.This essay received honorable mention in the 1991 competition of the Gravity Research Foundation.     

Anticancer agent development: X-ray crystal structure and keto-enol tautomerism of dimethyl 1-hydroxy-6,7-methylenedioxy-4-(3′,4′,5′-trimethoxyphenyl-trans-3,4-dihydronaphthalene-2,3-dicarboxylate

Structural and conformational information obtained from the crystal structure and solution¹H nmr investigations of the title compound are compared. The 4-aryltetralone, C₂₄H₂₄O₁₀, crystallizes as a chloroform solvate in the monoclinic space group, P2₁/n, witha=12.519(4),b=17.938(6),c=12.534(9)Å,=111.90(5)°, and D_calc=1.51 g cm^–3 forZ=4. The data for this compound were collected at –150°C. Least-squares refinement of 2796 observed [F _o5(F _o)] reflections led to the final agreement index ofR=0.062. A threefold static disorder was observed for one of the carboxyl groups. The second carboxyl group participates in an intramolecular hydrogen bond and is thus ordered. The¹H nmr spectrum revealed the title compound to exist as a keto-enol tautomeric mixture in solution. Vicinal hydrogen coupling constant analysis proved reliable in ascertaining B-ring stereochemistry of 2,3-disubstituted-4-aryltetralones.     

A prenyloxycoumarin from Psiadia dentata

A new coumarin identified as 5-hydroxy-6-methoxy-7-(3-methyl-but-2-enyloxy)-2H-1-benzopyran-2-one (isoobtusitin) was isolated from Psiadia dentata. This compound showed, in vitro, a moderate inhibitory activity against poliovirus and a very weak activity against (HIV), whereas it was inactive against (HSV1), (VSV), and murine tumoral cell lines (3LL, L1210).     

Isolement du prestegane B à partir de Steganotaenia araliacea hochst., premier lignane bis-(meta-hydroxy-benzyl)-butanolide d'origine végétale. - Isolement et synthèse en une étape du diméthylmatairésinol-

Prestegane B -the first higher plant occured dibenzyl butanolide lignan with two $\text{meta}$-phenols- and its dimethylether were isolated from $\text{S.araliacea}$ and the structure determinations were performed with the aid of high resolution NMR and Mass fragmentation for the former, and with 2-D J resolved ¹H NMR studies and synthesis for the latter.     

Two-photon spectra of gases by three-wave mixing

We show that the two-photon resonances in the third-order susceptibility can be exploited to yield two-photon spectra of molecular gases at moderately high spectral resolution. This form of spectroscopy does not depend on the occurrence of processes (such as fluorescence of photoionization) leading to indirect methods of two-photon absorption. The method is direct and leads in principle to values for a two-photon cross section. Comparisons of two-photon and coherent anti-Stokes Raman resonances leads to ratios of Raman and two-proton cross sections independent of the laser powers and spatial characteristics. The technique is documented with rotationally resolved spectra of SO₂ and NO. A value for |α_xxα_yy| of 1.5 × 10^?52 cm⁶ was measured for the O₁₂ (6$\text{built:1}\text{2}$) component of the (A)²Σ ← (X)² Π_{$\text{1}\text{2}$} transition of nitric oxide.     

The temperature dependence of the d-electron dipole strengths of cobalt(II) tetrahalides

The dipole strengths of the ⁴A₂ → ⁴T₁(F), ⁴T₁(P), d-electron transitions of the cobalt(II) tetrahalides are found to decrease with an increase in temperature, in agreement with a dynamic ligand-polarisation model for the absorption mechanism. The reduction in dipole strength is accounted for by a decrease in the coulombic coupling between the quadrupole moment of the d-electron transition of the metal ion and the transient electric dipole moment induced in the ligands, due to the anharmonic increase in the mean bond length and the progressively larger mean-square amplitudes of the bending modes of the complex ion in its ground electronic state as the temperature is raised.