The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

If G is a connected graph of order n ⩾ 1, then by a hamiltonian coloring of G we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| ⩾ n − 1 − D _G (x, y) (where D _G (x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ V (G). Let G be a connected graph. By the hamiltonian chromatic number of G we mean

Tribological behaviour of nanostructured Ti-C:H coatings for biomedical applications

The development of a mechanically stable, functionally graded Ti-doped a-C:H interface layer in combination with a functional a-C:H coating requires a reduction of the brittle phases which induce generally problems in the transitions from Ti to TiC/a-C:H. The core objective of this study was to develop an optimum interlayer between the substrate and the functional top layer for biomedical applications, namely for tooth implants. Since the interlayer may be exposed to the sliding process, in the case of local failure of the top layer it has to fulfil the same criteria: biocompatibility, high wear resistance and low friction.The functional Ti-C:H layers with thickness in the range 2.5–3.5 μm were deposited by a magnetron sputtering/PECVD hybrid process by sputtering a Ti-target in a C₂H₂ + Ar atmosphere in dc discharge regime. The sets of coating samples were prepared by varying the C and H concentrations controlled by the C₂H₂ flow during the deposition process. The tribological properties were evaluated on a pin-on-disc tribometer at room temperature (RT) and at 100 °C using 440C balls with a diameter of 6 mm. The tests at 100 °C were performed to investigate the effect of the sterilization temperature on the tribological properties and the coating lifetime as well. The tribological performance was examined with respect to the friction coefficient, the wear rates of the coating and the counter-parts and the analysis of the wear debris. The Ti/C ratio decreased almost linearly from 4.5 to 0.1 with increasing C₂H₂ flow; the hydrogen content showed a minimum of 5 at.% at C₂H₂ flow of 30 sccm, while for lower flows it was about 10 at.%. The coatings could be divided into three groups based on the C₂H₂ flow: (i) 10–15 sccm, exhibiting severe abrasive damage during the sliding tests, (ii) 20–45 sccm, showing the highest hardness and friction values, and (iii) 52–60 sccm, with moderate hardness and minimal values of the friction coefficient and the wear rate.     

Carbon Nanotube Electrodes for Hot‐Wire Electrochemistry

Hot‐wired electrodes : Thin metallic wires (d=25 μm) are covered with a 3 μm layer of single‐walled carbon nanotubes (SWCNTs; see image) by electrophoresis from a suspension containing excess ionic surfactant. A pure SWCNT surface is achieved by heating the electrode in air. Strong differences between covered and bare metallic electrodes occur with in situ heating during electrochemical experiments.

     

The Odd-Distance Plane Graph

The vertices of the odd-distance graph are the points of the plane ℝ². Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ² is countably choosable, while such a graph in ℝ³ is not. The research of J. Maňuch was supported in part by MITACS (Mathematics of Information Technology and Complex Systems). The research of M. Rosenfeld was supported in part by the Chancellor Research Grant and the Institute of Technology, UWT. The research of S. Shelah was supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by NSF grant No. NSF-DMS 0600940. No. 923 on Shelah’s publication list. The research of L. Stacho was supported in part by NSERC (Natural Science and Engineering Research Council of Canada) grant.     

A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph.     

Stereospecific Synthesis of D -Glycero- D -ido-oct-2-ulose

 D-Glycero-D-gulo-heptose reacted with 2,2-dimethoxypropane to give its 2,3:6,7-di-O-isopropylidene derivative. Its base-catalyzed addition to formaldehyde resulted in the formation of 2,3:6,7-di-O-isopropylidene-2-C-(hydroxymethyl)-D-glycero-D-gulo-heptofuranose. After acid hydrolysis of this aldolization product, a new branched-chain aldose, 2-C-(hydroxymethyl)-D-glycero-D-gulo-heptose, was obtained, which was stereospecifically rearranged under the catalytic action of molybdic acid to D-glycero-D-ido-oct-2-ulose.     

Study of β relaxation in polypropylene fibres modified by several techniques

Polypropylene fibres prepared from polypropylene (PP) blended with 8 wt% of an ethylene aminoalkylacrylate copolymer were studied by means of ¹H nuclear magnetic resonance (NMR) and dynamic mechanical techniques. Measurements were carried out on the fibres spun at the spinning speeds 400 and 1200 m.min⁻¹ and then drawn to the draw ratio λ = 3.5. Spin ‐ lattice relaxation time T₁ and storage E' and loss E'' components of a dynamic mechanical modulus were measured in the temperature region of the glass transition. Broad line NMR spectra were detected at 313 K. The experimental results show that drawing of the fibres prepared at both spinning speeds results in a restriction of the molecular motion. Simultaneously with the restriction, a chain mobility enhancement was observed in noncrystalline regions of the fibres. Similar effects were induced also by rising the spinning speed. Reasons for this behavior are discussed in the paper.     

Three-dimensional lattice ground states for Riesz and Lennard-Jones–type energies

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones–type potentials $f_{n,m}^{\mathrm{LJ}}(r):=a{r}^{-n}-b{r}^{-m}$, $n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the body-centered-cubic (BCC) lattice, face-centered-cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close-packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for and the HCP lattice is computed to have higher energy than the FCC (for $s>3/2$) and BCC (for ) lattices. In the Lennard-Jones case with exponents , the ground state among lattices is confirmed to be an FCC lattice whereas an HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type “FCC-SH” and “FCC-HCP-SH” (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents $(n,m)$. In the SH phase, the variation of the ratio Δ between the interlayer distance d and the lattice parameter a is studied as V increases. In the critical region of exponents , the SH phase with an extreme value of the anisotropy parameter Δ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For , the BCC lattice is the only energy minimizer. Choosing , the FCC and SH latices become minimizers.