Bandwidth of the strong product of two connected graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)-f(y)|:xy∈E(G)} taken over all proper numberings f of G. The strong product of two graphs G and H, written as G(S_P)H, is the graph with vertex set V(G)×V(H) and with (u₁,v₁) adjacent to (u₂,v₂) if one of the following holds: (a) u₁ and v₁ are adjacent to u₂ and v₂ in G and H, respectively, (b) u₁ is adjacent to u₂ in G and v₁=v₂, or (c) u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the strong product of two connected graphs. Let G be a connected graph. We denote the diameter of G by D(G). Let d be a positive integer and let x,y be two vertices of G. Let denote the set of vertices v so that the distance between x and v in G is at most d. We define δ_d(G) as the minimum value of over all vertices x of G. Let denote the set of vertices z such that the distance between x and z in G is at most d-1 and z is adjacent to y. We denote the larger of and by . We define η(G)=1 if G is complete and η(G) as the minimum of over all pair of vertices x,y of G otherwise. Let G and H be two connected graphs. Among other results, we prove that if δ_D(H)(G)?B(G)D(H)+1 and B(H)=⌈(|V(H)|+η(H)-2)/D(H)⌉, then B(G(S_P)H)=B(G)|V(H)|+B(H). Moreover, we show that this result determines the bandwidth of the strong product of some classes of graphs. Furthermore, we study the bandwidth of the strong product of power of paths with complete bipartite graphs.     

Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

This Note is devoted to the study of the Fano manifolds X obtained by blow-up along a smooth curve C in a complex projective manifold Y. By the Mori theory, we can ensure the existence of an extremal contraction $\phi :X\to Z$ different from the blow-up $\pi :X\to Y$. Here we give the complete classification of the corresponding pairs $(Y,C)$ in the case where φ is a fiber type contraction of relative dimension 2, i.e. the general fibers of φ are del Pezzo surfaces. In Tsukioka (Thesis, Nancy University 1, 2005), the relative dimension 1 case is also considered. To cite this article: T. Tsukioka, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A color-coded backlighted defocusing digital particle image velocimetry system

Defocusing digital particle image velocimetry (DDPIV), as a true three-dimensional (3D) measurement technique, allows for the measurement of 3D velocities within a volume. Initially designed using a single CCD and 3-pinhole mask (Willert and Gharib in Exp Fluids 12:353–358, 1992), it has evolved into a multi-camera system in order to overcome the limitations of image saturation due to multiple exposures of each particle. In order to still use a single camera and overcome this limitation, we have modified the original single CCD implementation by placing different color filters over each pinhole, thus color-coding each pinhole exposure, and using a 3-CCD color camera for image acquisition. Due to the pinhole mask, there exists the problem of a significant lack of illumination in a conventional lighting setup, which we have solved by backlighting the field-of-view and seeding the flow with black particles. This produces images with a white background superimposed with colored triple exposures of each particle. A color space linear transformation is used to allow for accurate identification of each pinhole exposure when the color filters’ spectrum does not match those of the 3-CCD color camera. Because the imaging is performed with a multi-element lens instead of a single-element lens, an effective pinhole separation, d _e, is defined when using a pinhole mask within a multi-element lens. Calibration results of the system with and without fluid are performed and compared, and a correction of the effective pinhole separation, d _e, due to refraction through multiple surfaces is proposed. Uncertainty analyses are also performed, and the technique is successfully applied to a buoyancy-driven flow, where a 3D velocity field is extracted.     

Photomobile Polymer Materials with Crosslinked Liquid‐Crystalline Structures: Molecular Design,Fabrication, and Functions

Crosslinked liquid‐crystalline polymer materials that macroscopically deform when irradiated with light have been extensively studied in the past decade because of their potential in various applications, such as microactuators and microfluidic devices. The basic motions of these materials are contraction–expansion and bending–unbending, which are observed mainly in polysiloxanes and polyacrylates that contain photochromic moieties. Other sophisticated motions such as twisting, oscillation, rotation, and translational motion have also been achieved. In recent years, efforts have been made to improve the photoresponsive and mechanical properties of this novel class of materials through the modification of molecular structures, development of new fabrication methods, and construction of composite structures. Herein, we review structures, functions, and working mechanisms of photomobile materials and recent advances in this field.     

Steady-state polarization measurements of lithium insertion electrodes for high-power lithium-ion batteries

Steady-state polarization measurements of lithium titanium oxide (LTO; Li[Li_1/3Ti_5/3]O₄) were carried out using the 0-V lithium-ion cells consisting of two identical LTO-electrodes with a parallel-plate symmetrical electrode configuration. The sinusoidal voltage with the peak amplitude of 1.0 V was imposed at 0.1 Hz upon the 0-V cells and the current response was measured as a function of time. The steady-state polarization, obtained by plotting the current versus applied voltage, was linear in current up to approximately 60 mA cm^?2 or 4 A g^?1 based on the LTO weight and suggested the resistance polarization only for the lithium insertion electrode of the LTO. The method was also applied to lithium aluminum manganese oxide (LAMO; Li[Li_0.1Al_0.1Mn_1.8]O₄) and the resistance polarization of the LAMO-electrode was determined for currents up to approximately 25 mA cm^?2 or 2 A g^?1 based on the LAMO weight. The validity of the results was examined for the polarization measurements of the 2.5-V lithium-ion battery consisting of LTO and LAMO, and the significance of the polarization measurements of lithium insertion electrodes for high-power applications was discussed.     

Exploring the thermochromism of sulfite-embedded polyoxometalate capsules

The Dawson-type polyanion [α-Mo(18)O(54)(SO(3))(2)](4-), with two SO(3)(2-) templates embedded inside a polyoxomolybdate(vi) cage, exhibits thermochromism over an exceptionally wide temperature range (～500 K). The temperature dependence of the cluster structure, established from X-ray crystallography, IR and Raman spectroscopy and DFT calculations, is related to a decreasing HOMO-LUMO gap in the near UV with increasing temperature. We postulate this is due to geometrical changes that affect both the occupied and unoccupied frontier molecular orbitals of this cluster anion.     

Theoretical study on singlet oxygen adsorption onto surface of graphene-like aromatic hydrocarbon molecules

Recently, graphene sheet is one of interesting systems to realize novel electronic properties. Especially, interaction between graphene and adsorbed oxygen molecule is very important to control electronic condition. In this paper, we employed some aromatic hydrocarbons as simple systems of graphene sheet and ab initio MO calculations were carried out to investigate inter-molecular interaction. It is found that not triplet but singlet O₂ molecule have potential of chemisorption onto graphene surface. From the calculated potential energy surface (PES) for distance between benzene and O₂ molecules, meta-stable structure is found at about 1.5 Å with potential barrier. In the optimized structure of its meta-stable state, structural strain can be relaxed through bending of planer benzene ring. Its energy is estimated at 70.10 kcal/mol for benzene. We also estimated the strain effects for naphthalene and pyrene molecules as larger case of graphene and they were 80.85 and 72.45 kcal/mol, respectively.     

Hemin-adsorbed carbon felt for sensitive and rapid flow-amperometric detection of dissolved oxygen

Hemin was physically adsorbed onto porous carbon felt (CF), a microelectrode ensemble of micro-carbon fiber (ca. 7 μm in diameter) and possessing a three-dimensional random structure. The hemin-CF exhibited a well-defined redox wave that is due to Fe(III)/Fe(II) redox process in hemin, with a formal potential of ?0.32 V (vs. Ag/AgCl) in deoxygenated buffer solution of pH 7.0. The surface coverage of the electroactive hemin molecules on the surface of the CF was calculated to be 5.0?×?10^?11 mol cm^?2, and the apparent heterogeneous electron transfer rate constant is 3.35 s^?1. The hemin-CF electrode displays excellent electrocatalytic activity for the reduction of dissolved oxygen (DO), and the magnitude of the cathodic current increases with increasing concentrations of DO in the sample solution. The electrode was used as a flow-through detector for sensitive and rapid consecutive determination of DO. Deoxygenated pH 7.0 solutions were analyzed at a flow rate of 8.0 mL min^?1 at an applied potential of ?0.2 V, and highly reproducible cathodic peak current responses to DO were observed in the 0.72 to 13.3 mg L^?1 concentration range. The maximum throughput is 170 samples h^?1. The hemin-CF-based amperometric flow-sensor was applied to determine the concentration of DO in environmental water samples.