Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups

We apply a construction of Rips to show that a number of algorithmicproblems concerning certain small cancellation groups and, inparticular, word hyperbolic groups, are recursively unsolvable.Given any integer k > 2, there is no algorithm to determinewhether or not any small cancellation group can be generatedby either two elements or more than k elements. There is a smallcancellation group E such that there is no algorithm to determinewhether or not any finitely generated subgroup of E is all ofE, or is finitely presented, or has a finitely generated secondintegral homology group.     

On the maslov index

In this paper we give four definitions of Maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other. Moreover, relationships of several symplectic and differential geometric, analytic, and topological invariants (including triple Maslov indices, eta invariants, spectral flow and signatures of quadratic forms) to the Maslov index are developed and formulae relating them are given. The broad presentation is designed with a view to applications both in geometry and in analysis. © 1994 John Wiley & Sons, Inc.     

Anomalous optical response with variation in band filling in linear chain conductors

We report the polarized infrared reflectance of ($\text{N}$-methylphenazinium)_x (phenazine)_1-x(tetracyanoquinodimethanide), (NMP)_x(Phen)_1-x(TCNQ) for 0.5≤x≤1.0. We demonstrate that the “Drude edge” in TCNQ salts is related to the conduction electron charge density. An anomalous rapid red shift of the plasma frequency, observed for $\text{x ～}\text{2}\text{3}$, may signal the emptying of the conduction electrons from half of the chains and/or a crossover from small U/W (coulomb repulsion/bandwidth) to large U/W behavior.     

Ultrasonic detection of resonant cavitation bubbles in a flow tube by their second-harmonic emissions

During low-power exposures, biophysical effects of ultrasonic cavitation are induced primarily by resonant bubbles, and there is a need for a new method of detecting these small bubbles. Bubble pulsation theory indicates that second-harmonic emissions emanate from resonant bubbles even at low amplitudes. A device was constructed to detect resonant bubbles passing through it in a flowing liquid by monitoring second-harmonic responses to a low amplitude, 1.64 MHz ultrasonic field. During testing, 4.2 μm diameter resonant bubbles produced signals 40 times larger than 500 μm diameter bubbles, and this technique was much better than a first-harmonic scattering technique for counting resonant bubbles.     

Non-linear behaviour and stability of trapped micron-sized cylindrical gas bubbles in an ultrasonic field

Stable arrays of cylindrical bubbles with diameters 3 and 4 μm were produced by trapping air in the pores of hydrophobic Nuclepore filters. These bubble arrays were irradiated by beams of cw and pulsed ultrasound at carrier frequencies near the resonance frequencies of the trapped bubbles. By examining the frequency spectrum of the field scattered by the arrays it was found that the bubbles may oscillate as non-linear stable cavities of long duration. Using short pulses, the bubbles remain stable up to excitation pressure amplitudes of several bars. Mechanisms for the eventual growth and decay of the bubbles are discussed in light of the observations.     

Two-dimesional electromagnetic fields guided by plasma layers. II

Radiophysics and Quantum Electronics -     

Quantum insights from null-strut geometrodynamics

We discuss quantum insights due to the null-strut formalism. These insights deal primarily with two topics: the formalism of a theory of canonical simplicial quantum gravity based on the geometrodynamic duality of null-strut calculus, and the natural implementation of spinors and spin networks in null-strut calculus.