Generalization of a formula of Touchard for Catalan numbers

In this paper a general identity is proved which includes among its special cases a recurrence formula for Catalan numbers given by J. Touchard in 1924.     

Frequency pulling and heterodyning observed in voltage spectrum of the Fröhlich conductor NbSe3

The voltage power spectrum of NbSe₃ has been studied in the upper charge-density wave (CDW) state where only one CDW exists. In the non-Ohmic regime sharp spikes appear in the spectrum. The frequency of these spikes increases monotonically with current as in the lower CDW state. We study the “fine structure” of the spectrum and find evidence for “frequency pulling” between the modes. A linear coupled-mode treatment is used to model this effect. There is also evidence of heterodyning of an intrinsic (current independent) 2MHz mode. We briefly discuss these phenomena in terms of Fröhlich sliding conductivity. Difficulties with the simple two-fluid model as well as models based on solid state turbulence are pointed out.     

The effects of filamentary conduction through uniform and flawed filaments in insulators and semiconductors

Electrical conduction through relatively low resistance filaments has been proposed as the mechanism whereby current-controlled negative resistance (CCNR), voltage-controlled negative resistance (VCNR), switching and memory effects take place, particularly in amorphous insulators and semiconductors. Two types of filament are considered: uniform filaments (in which the applied voltage is assumed to drop uniformly along the filament's length) and non-uniform filaments (in which the applied voltage drop occurs primarily at a higher resistance flaw within the filament). By regarding heat dissipation as taking place either uniformly or preferentially at flaws, depending on the filament type, expressions are derived for the temperature within the non-conducting insulator or semiconductor regions, filament or flaw temperature, and voltage at which a filament will cease to conduct. It is shown that for uniform filamentary conduction in thin films the conduction in individual filaments is influenced by the number and proximity of neighbouring filaments, while flawed filaments are essentially isolated from their neighbours. An illustration in the application of the model is given (in the case of uniform filaments) for an assembly of conducting filaments, and various filamentary parameters are calculated.Experimental evidence for a range of different systems is reviewed, and shown to be consistent with the model.     

A note on graphs whose neighborhoods are n-cycles

Let G be a graph, and let v be a vertex of G. We denote by N(v) the set of vertices of G which are adjacent to v, and by 〈N(v)〉 the subgraph of G induced by N(v). We call 〈N(v)〉 the neighborhood of v. In a paper of 1968, Agakishieva has, as one of her main theorems, the statement: “Graphs in which every neighborhood is an n-cycle exist if and only if 3?n?6.” It it is the object of this note to provide a list of counter examples to this statement.     

Minimal Degree and (k, m)-Pancyclic Ordered Graphs

Given positive integers k m n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.     

Graph minors and linkages

Bollobás and Thomason showed that every 22k‐connected graph is k‐linked. Their result used a dense graph minor. In this paper, we investigate the ties between small graph minors and linkages. In particular, we show that a 6‐connected graph with a K minor is 3‐linked. Further, we show that a 7‐connected graph with a K minor is (2,5)‐linked. Finally, we show that a graph of order n and size at least 7n?29 contains a K minor. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 75–91, 2005     

The Size and Composition of a Road Transport Fleet

The paper starts with a discussion of the simple fleet size problem. It is shown that this simple problem can be formulated as a linear program.The second part of the paper consists of an actual case study. The fleet concerned is faced with highly seasonal demand which can be met by the firm's own vehicles or by outside hire. There are two types of vehicle, both of which are available in six different sizes. Linear programming was used to find the optimum size and composition of the company fleet. The results, which were substantially implemented, recommended a smaller company fleet and concentration on larger and more flexible vehicles.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.