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.     

FFT solution of the Robbins problem

An FFT method is described for the solution of Poisson's equationover a rectangular region with Robbins boundary conditions oneither one or two sides of the region, together with suitableconditions on the rest of the boundary. In contrast to earlierapplications of the FFT technique, the equations for the Fourierharmonic amplitudes do not decouple into simpler independentsystems and an effective iterative scheme is developed for thesolution of these equations. A theoretical convergence analysisis shown generally to support the results obtained from practicalcomputation. For the test problems considered the method isfound to take between 3 and 4 times the execution time for problemssoluble directly by the FFT technique.     

A method for investigating the orientational behaviour of fibrous particles in gaseous flow

A method is described by which the angular orientation distribution of fibrous particles carried in a gaseous stream may be investigated. The method is based upon the interpretation of the spatial intensity distribution or scattering profile of laser light scattered by individual fibres. The scattering instrument used to capture the profiles is described, and the mathematical computation required to ascertain the orientation of each particle at the measurement point is detailed. Illustrative results are given for a study of airborne micromachined silicon particles of 12 μm length and 1.0 μm by 1.5 μm cross-section. The method is currently being employed by the authors to investigate ways of improving the orientation control over nonspherical particles in systems such as aerodynamic particle sizers and particle shape classifiers, since lack of particle orientation control is known to adversely affect the measurement accuracy of both these types of instrument.     

Isomer effects on transport properties of polyesters based on bisphenol-A

Pure gas sorption and transport properties of polyesters based on bisphenol-A and both pure isophthalic and pure terephthalic acid chloride were obtained for He, N₂, O₂, CH₄, and CO₂ at 35°C. The polymers were synthesized in our laboratory and amorphous films were prepared with a specialized solvent casting procedure. The polymer containing m-phenylene groups shows higher permselectivity for most of the gas pairs. The ideal selectivity of O₂/N₂ was increased by 33% when p-phenylene units were replaced by m-phenylene ones. On the other hand, the polyester containing only p-phenylene groups, shows higher permeability to all the gases studied. The polymer based on pure terephthalic acid chloride has a 75% higher oxygen permeability and a 1.1-fold higher carbon dioxide permeability than the isophthalic acid derivative. The polyester containing meta-phenylene units has lower T_g, higher permselectivity, lower permeability, lower fractional free volume (FFV), and lower d-spacing. The values of FFV, and lower d-spacing. The values of FFV and d-spacing were only slightly different between the two isomers. Moreover, for the sub-T_gγ transition the maximum in tan δ occured at essentially the same temperature (?55°C). The polymer with a higher concentration of p-phenylene units shows somewhat larger area under the γ-peak, indicating slightly more sub-T_g motion. The Distribution of FFV is considered to be the determining factor for the differences in transport properties observed. © 1993 John Wiley & Sons, Inc.     

Logarithmic corrections for dilute uniaxial ferromagnets at the critical dimension

The leading logarithmic corrections to the critical behavior of a dilute uniaxial (Ising) ferromagnet in the disordered phase are derived using renormalization group methods. The values of the exponents in the logarithmic terms differ from those given by previous authors.     

Periodic waves with constant vorticity in water of infinite depth

Periodic waves propagating at a constant velocity at the surfaceof a fluid with constant vorticity in water of infinite depthare considered. The problem is solved numerically by a boundary-integral-equationmethod. Simmen & Saffman (Stud. Appl. Maths 75, 35, 1985)showed that there are families of solutions which have limitingconfigurations with a 120 angle at their crests or a trappedbubble at their troughs. It is shown that there are additionalfamilies of solutions. These families have limiting configurationswith trapped bubbles at their crests. Each bubble is circularand contains fluid in rigid-body rotation. The results are consistentwith previous calculations for solitary waves in water of finitedepth.     

Selective spatial localization of the atom displacements in one-dimensional hybrid quasi-regular (Thue-Morse and Rudin-Shapiro)/periodic structures

We have studied the vibrational frequencies and atom displacements of one-dimensional systems formed by combinations of Thue-Morse and Rudin-Shapiro quasi-regular stackings with periodic ones. The materials are described by nearest-neighbor force constants and the corresponding atom masses. These systems exhibit differences in the frequency spectrum as compared to the original simple quasi-regular generations and periodic structures. The most important feature is the presence of separate confinement of the atom displacements in one of the parts forming the total composite structure for different frequency ranges, thus acting as a kind of phononic cavity.     

A connection between numerical ranges of selfadjoint matrix polynomials

In this paper, a geometric connection between the joint numerical range and the numerical range for selfadjoint matrix polynomial is presented. Moreover, algebraic properties for these ranges are investigated.     

A shape model for the twisting power of chiral solutes in nematics

The twisting power of chiral probes in nematics is interpreted in terms of a shape model, in which the surface elements of the solute molecules tend to align with the local director. The theoretical treatment is based on a previous approach, suitable for relating order parameters in nematics to molecular shape, and leads to the definition of a molecular pseudo-tensor whose orientational average determines the pitch and handedness of the helical macrostructures. Results of numerical calculations performed for distorted biphenyl and binaphthyl molecules are in agreement with the experimental results.