Hamiltonian circuits in Cayley graphs

The following result is proved: If either G is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected Cayley graph of G is hamiltonian.     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Unexpected density fluctuations in jammed disordered sphere packings

We computationally study jammed disordered hard-sphere packings as large as a million particles. We show that the packings are saturated and hyperuniform, i.e., that local density fluctuations grow only as a logarithmically augmented surface area rather than the volume of the window. The structure factor shows an unusual nonanalytic linear dependence near the origin, S(k) approximately |k|. In addition to exponentially damped oscillations seen in liquids, this implies a weak power-law tail in the total correlation function, h(r) approximately -r(-4), and a long-ranged direct correlation function c(r).     

Hole interactions with molecular vibrations on DNA

We report on a study of the interactions between holes and molecular vibrations on dry DNA using photoinduced infrared absorption spectroscopy. Laser photoexcited holes are found to have a room-temperature lifetime in excess of tau > 1 ms, clearly indicating the presence of localization. However, from a quantitative model analysis of the frequency shifts of vibrational modes caused by the holes, we find the hole-vibrational coupling constant to be relatively small, lambda approximately 0.2. This interaction leads to a change in the conformational energy of DeltaE0 approximately 0.015 eV, which is too small to cause self-trapping at room temperature. We conclude that, at least in the dry (A) form, DNA is best understood in terms of a double chain of coupled quantum dots arising from the pseudorandom chain sequence of base pairs, in which Anderson localization prevents the formation of a metallic state.     

Phosphorus as sintering activator in powder metallurgical steels: characterization of the distribution and its technological impact

Powder metallurgy is a highly developed method of manufacturing reliable ferrous parts. The main processing steps in a powder metallurgical line are pressing and sintering. Sintering can be strongly enhanced by the formation of a liquid phase during the sintering process when using phosphorus as sintering activator. In this work the distribution (effect) of phosphorus was investigated by means of secondary ion mass spectrometry (SIMS) supported by Auger electron spectroscopy (AES) and electron probe micro analysis (EPMA). To verify the influence of the process conditions (phosphorus content, sintering atmosphere, time) on the mechanical properties, additional measurements of the microstructure (pore shape) and of impact energy were performed. Analysis of fracture surfaces was performed by means of scanning electron microscopy (SEM). The concentration of phosphorus differs in the samples from 0 to 1% (w/w). Samples with higher phosphorus concentrations (1% (w/w) and above) are also measurable by EPMA, whereas the distributions of P at technically relevant concentrations and the distribution of possible impurities are only detectable (visible) by means of SIMS. The influence of the sintering time on the phosphorus distribution will be demonstrated. In addition the grain boundary segregation of P was measured by AES at the surface of in-situ broken samples. It will be shown that the distribution of phosphorus depends also on the concentration of carbon in the samples.     

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

Nonexistence of some Antipodal Distance-regular Graphs of Diameter Four

We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri i , Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .     

Tight Distance-Regular Graphs

We consider a distance-regular graph with diameter d 3 and eigenvalues k = ₀ > ₁ > ... > _d . We show the intersection numbers a ₁, b ₁ satisfy

Groups acting on quasiconvex spaces and translation numbers

We prove that groups acting geometrically on -quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero.