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.     

Gas-phase electron diffraction determination of the molecular structure of perfluoro(methyloxirane)

The molecular geometry of perfluoro(methyloxirane) has been studied using gas-phase electron diffraction data, effective least-squares refinement of the structure being achieved with the aid of constraints to limit the number of variable parameters. With the CCF₃ bond constrained to be 0.078 Å longer than the ring CC, the refined bond- length values CF (av.) = 1.323(2), CO (av.) = 1.410(8), and CC (ring) = 1.467(7) Å ($\text{r}$_g values, with e.s.d. in parentheses) were obtained; the angles between ring bonds and substituent CF bonds were CCF (av.) 121(1) and OCF (av.) 114(1)^o, the corresponding parameters involving the bulkier CCF₃ fragment being larger by 3^o in each case [∠CCCF₃ 124(1)^o∠OCCF₃ 117(2)^o]. The remaining refined parameters were ∠CCF(of CF₃) = 110.6(4)^o and τ , a torsion angle defining the orientation of the CF bonds of the CF₃ group with respect to ring bonds, = 29(2)^o. Dependent bond angles possessed the values 62.7 (COC), 58.7 [OCC (ring)], 108.3 [FCF (CF₃ group)], 114 [FCF (ring CF₂)], and 111^o (FCCF₃).     

Synthesis of pyridone and pyridine rings by [4+2] hetero-cyclocondensation

Substituted pyridones and pyridines have been synthesised efficiently by employing iminium salt as a key precursor. These compounds were prepared using tandem [4+2] cycloaddition/deamination between azabutadiene and dienophiles.     

Colloidal nano-MOFs nucleate and stabilize ultra-small quantum dots of lead bromide perovskites

The development of synthetic routes to access stable, ultra-small (i.e. <5 nm) lead halide perovskite (LHP) quantum dots (QDs) is of fundamental and technological interest. The considerable challenges include the high solubility of the ionic LHPs in polar solvents and aggregation to form larger particles. Here, we demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites. Cr₃O(OH)(H₂O)₂(terephthalate)₃ (Cr-MIL-101), made of large mesopore-sized pseudo-spherical cages, allows fast and efficient diffusion of perovskite precursors within its pores, and promotes the formation of stable, ∼3 nm-wide lead bromide perovskite QDs. CsPbBr₃, MAPbBr₃ (MA⁺ = methylammonium), and (FA)PbBr₃ (FA⁺ = formamidinium) QDs exhibit significantly blue-shifted emission maxima at 440 nm, 446 nm, and 450 nm, respectively, as expected for strongly confined perovskite QDs. Optical characterization and composite modelling confirm that the APbBr₃ (A = Cs, MA, FA) QDs owe their stability within the MIL-101 nanocrystals to both short- and long-range interfacial interactions with the MOF pore walls.

We demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites.     

Physicochemical and structural properties of bacteriostatic sulfonamides: Theoretical study

A theoretical study at the Hartree–Fock and density functional theory levels is performed on sulfonamide‐type bacteriostatic compounds with the aim to provide an insight into their structure–activity relationship. The basicity of the p‐amino group is analyzed by means of the proton affinities and the protonation energies, showing that molecules presenting bacteriostatic activity are less basic, i.e., they are characterized by larger protonation energies and smaller proton affinities. The acidity of the amide group is analyzed through the deprotonation energy. The results reveal that the more acidic molecules present a larger bacteriostatic activity. This result is also confirmed from a study of bond orders. A bond order analysis of the amide group suggests that the electron attracting group in these molecules is responsible for the increase in acidity. The charge of the SO₂ group is also shown to be affected by the presence of the electron attracting group and consequently related to the acidity of the molecules. A geometric analysis shows that structures in which the amino group is more coplanar with respect to the benzenic ring possess larger bacteriostatic activity. A conformational analysis of these molecules illustrates that active molecules have relatively larger torsion energy barriers. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 165–172, 2003