On the Girth of Graphs Critical with Respect to Edge-Colourings

A graph G with maximum valency r is called critical if r + 1colours are needed for an edgecolouring, but every proper subgraphrequires at most r. In this note we consider the minimum orderf(r, g) of a critical graph of maximum valency r and girth g.We show that f(r, 3) = r+1 or r+2 according as r is even orodd, f(r, 4) = 2r+1,f(3, 5) = 9 and f(3, 6) = 15.     

Liquid polyamorphism: Possible relation to the anomalous behaviour of water

We present evidence from experiments and computer simulations supporting the hypothesis that water displays polyamorphism, i.e., water separates into two distinct liquid phases. This concept of a new liquid-liquid phase transition is finding application to other liquids as well as water, such as silicon and silica. Specifically, we investigate, the relation between changes in dynamic and thermodynamic anomalies arising from the presence of the liquid-liquid critical point in (i) Two models of water, TIP5P and ST2, which display a first order liquid-liquid phase transition at low temperatures; (ii) the Jagla model, a spherically symmetric two-scale potential known to possess a liquid-liquid critical point, in which the competition between two liquid structures is generated by repulsive and attractive ramp interactions; and (iii) A Hamiltonian model of water where the idea of two length/energy scales is built in. This model also displays a first order liquid-liquid phase transition at low temperatures besides the first order liquid-gas phase transition at high temperatures. We find a correlation between the dynamic fragility crossover and the locus of specific heat maxima C_P^max (“Widom line”) emanating from the critical point. Our findings are consistent with a possible relation between the previously hypothesised liquid-liquid phase transition and the transition in the dynamics recently observed in neutron scattering experiments on confined water. More generally, we argue that this connection between C_P^max and the dynamic crossover is not limited to the case of water, a hydrogen bonded network liquid, but is a more general feature of crossing the Widom line, an extension of the first-order coexistence line in the supercritical region. Dedicated to Armin Bunde on the occasion of his 60th birthday.     

The effect of ultrasonic vibrations on crystallization processes

Ultrasound has long been used during a phase transition to produce small, uniform crystals. Many theories have been suggested to explain the action of ultrasound in this application. This paper reviews the proposed mechanisms of ultrasonic crystallization.     

Statistical evaluation of methods for quantifying gene expression by autoradiography in histological sections

Background  

In situ hybridisation (ISH) combined with autoradiography is a standard method of measuring the amount of gene expression in histological sections, but the methods used to quantify gene expression in the resulting digital images vary greatly between studies and can potentially give conflicting results.     

Laser writing techniques for photomask fabrication using a femtosecond laser

Photomasks are the backbone of microfabrication industries. Currently they are fabricated by a lithographic process, which is very expensive and time consuming since it is a multi-step process. These issues can be addressed by fabricating photomasks by direct femtosecond laser writing, which is a single-step process and comparatively cheaper and faster than lithography. In this paper we discuss our investigations on the effect of two types of laser writing techniques, namely front- and rear-side laser writing, with regard to the feature size and the edge quality of a feature. It is proved conclusively that for the patterning of masks, front-side laser writing is a better technique than rear-side laser writing with regard to smaller feature size and better edge quality. Moreover the energy required for front-side laser writing is considerably lower than that for rear-side laser writing. Received: 22 May 2001 / Accepted: 14 September 2001 / Published online: 17 October 2001     

Multi-sample cluster analysis using Akaike's Information Criterion

Summary Multi-sample cluster analysis, the problem of grouping samples, is studied from an information-theoretic viewpoint via Akaike's Information Criterion (AIC). This criterion combines the maximum value of the likelihood with the number of parameters used in achieving that value. The multi-sample cluster problem is defined, and AIC is developed for this problem. The form of AIC is derived in both the multivariate analysis of variance (MANOVA) model and in the multivariate model with varying mean vectors and variance-covariance matrices. Numerical examples are presented for AIC and another criterion calledw-square. The results demonstrate the utility of AIC in identifying the best clustering alternatives. This research was supported by Office of Naval Research Contract N00014-80-C-0408, Task NR042-443 and Army Research Office Contract DAAG 29-82-K-0155, at the University of Illinois at Chicago.     

3D composition of epitaxial nanocrystals by anomalous X-ray diffraction: observation of a Si-rich core in Ge domes on Si(100)

Three-dimensional composition maps of nominally pure Ge domes grown on Si(001) at 600 degrees C were obtained from grazing incidence anomalous x-ray scattering data at the Ge K edge. The data were analyzed in terms of a stack of layers with laterally varying concentration. The results demonstrated that the domes contained a Si-rich core covered by a Ge-rich shell and were independently supported by selective etch experiments. The composition profile resulted from substrate Si alloying into the Ge during growth to partially relax the stress in and under the domes.