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.     

Experimental investigation of the aquatic propulsion caused by localised flexural wave propagation in immersed wedges and plates

This paper describes the results of the experimental verification of the idea of wave-like aquatic propulsion of autonomous and man-inhabited vessels first published about 10 years ago by one of the present authors (V.V.K.). The idea is based on employing the unique type of localised flexural elastic waves propagating along edges of wedge-like structures immersed in water (wedge elastic waves). Such wedge-like structures supporting localised elastic waves can be attached like fish fins to a body of a small ship or a submarine and used for their propulsion. The proposed principle of employing localised flexural waves as a source of aquatic propulsion has been biologically inspired by the specific swimming mode used in nature by stingrays. To verify the idea experimentally, the first working prototype of a small catamaran using the above-mentioned wave-like propulsion via the attached rubber keel has been build and tested in Loughborough University. The tests have been carried out in two phases, in a water tank and then in open water. The test results have shown that the catamaran was propelled very efficiently and could achieve the speed of 36 cm/s, i.e., about one vehicle length per second, thus demonstrating that the idea of wave-like propulsion of small man-inhabited craft is viable. The reported proof of the viability of this idea may open new opportunities for marine craft propulsion, which can have far reaching implications.     

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.     

Positive ion fragmentation mechanisms of some simple aldehydes and epoxides and trimethylene oxide

The metastable ion spectra of ethylene oxide, acetaldehyde, acetaldehyde-d₄, propylene oxide, propionaldehyde and trimethylene oxide are presented. They reveal no reason to suppose that electron-impact-induced isomerisation of oxide to aldehyde structure occurs, although many spectral features are common to various members of this set. The spectra of phenylacetaldehyde and styrene oxide are compared, and the more plausible possibility of partial isomerisation in this case is discussed.