Overlap in consistent cycles

A (directed) cycle C in a graph Γ is called consistent provided there exists an automorphism of Γ, acting as a 1‐step rotation of C. A beautiful but not well‐known result of J.H. Conway states that if Γ is arc‐transitive and has valence d, then there are precisely d ? 1 orbits of consistent cycles under the action of Aut(Γ). In this paper, we extend the definition of consistent cycles to those which admit a k‐step rotation, and call them ‐consistent. We investigate ‐consistent cycles in view of their overlap. This provides a simple proof of the original Conway's theorem, as well as a generalization to orbits of ‐consistent cycles. A set of illuminating examples are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 55–71, 2007     

Diverse coordination of two ligands in ferromagnetic [Cu(μ-HCO2)2(3-pyOH)]n and [Cu2(μ-HCO2)2(μ-3-pyOH)2(3-pyOH)2(HCO2)2]n

Two novel polynuclear complexes with methanoate anions and 3-hydroxypyridine ligands [Cu(μ-HCO₂)₂(3-pyOH)]_n (1) and [Cu₂(μ-HCO₂)₂(μ-3-pyOH)₂(3-pyOH)₂(HCO₂)₂]_n (2), respectively, were synthesized and characterized. The central copper atom in 1 is surrounded by four methanoates and a 3-pyOH molecule, forming a square-pyramidal CuO₃NO chromophore. All the methanoates are bidentate and serve as bridges between the adjacent copper ions via syn-anti and anti–anti coordination. The basal square coordination axes are formed by O(syn), N(3-pyOH) (1.974(2), 2.016(2) Å) and O(anti), O(anti) (1.945(2), 1.960(2) Å), while the third O(anti) (2.247(2) Å) is on the top of the pyramid. A ferromagnetic transition with an exchange constant 2J/k_B = 9.2 cm⁻¹ is found for 1 below 20 K. This interaction probably takes place through two syn-anti methanoates extended in a chain through the 2D structure. On the other hand, two monoatomic Cu–O–Cu intra-dinuclear asymmetric (1.986(2), 2.415(2) Å) bridges of two methanoates in [Cu₂(HCO₂)₄(3-pyOH)₄] (2) are present. An elongated distorted octahedral coordination sphere around each copper(II) atom is completed by an additional monodentate terminal methanoate (1.975(2) Å), two N-coordinated 3-pyOH (2.005(2), 2.002(2) Å) and the third weakly O-coordinated 3-pyOH (2.732(2) Å). Although a shorter Cu?Cu distance is noticed in 2 than in 1 (4.690(1) Å 1, 3.442(1) Å 2), much weaker ferromagnetism is found in 2.     

Stability and Degradation Mechanisms of Copper-Based Catalysts for Electrochemical CO2 Reduction

To date, copper is the only monometallic catalyst that can electrochemically reduce CO₂ into high value and energy-dense products, such as hydrocarbons and alcohols. In recent years, great efforts have been directed towards understanding how its nanoscale structure affects activity and selectivity for the electrochemical CO₂ reduction reaction (CO₂RR). Furthermore, many attempts have been made to improve these two properties. Nevertheless, to advance towards applied systems, the stability of the catalysts during electrolysis is of great significance. This aspect, however, remains less investigated and discussed across the CO₂RR literature. In this Minireview, the recent progress on understanding the stability of copper-based catalysts is summarized, along with the very few proposed degradation mechanisms. Finally, our perspective on the topic is given.     

On the nullspace of arc-transitive graphs over finite fields

Let A be the adjacency matrix of a graph???. The nullity of A (that is, the dimension of the nullspace of?A), when viewed as a matrix over a field of prime characteristic?p, is called the p-nullity of ??. We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertex-transitive graph of order a power of p is zero, provided that the valency of the graph is coprime to?p.     

Existence of Solutions for Semilinear Nonlocal Elliptic Problems via a Bolzano Theorem

In this paper we deal with the existence of positive solutions for the following nonlocal type of problems $$\everymath{\displaystyle} \left\{ \begin{array}{l@{\quad}l} -\Delta u = \frac{\sigma}{( \int_{\varOmega} g(u)\, dx )^p} f(u) & \mbox{in}\ \varOmega, \\[3mm] u>0 & \mbox{in}\ \varOmega, \\[1mm] u=0 & \mbox{on}\ \partial\varOmega, \end{array} \right. $$ where Ω is a bounded smooth domain in ? ^N (N≥1), f,g are continuous positive functions, σ>0 and p∈?. We give sufficient conditions on the functions f and g in order to have existence of positive solutions.     

In vitro photodynamic inactivation of Cryptococcus neoformans melanized cells with chloroaluminum phthalocyanine nanoemulsion

The selection of fungi resistant to currently used fungicides and the emergence of new pathogenic species make the development of alternative fungus-control techniques highly desirable. Photodynamic antimicrobial chemotherapy (PACT) is a promising method which combines a nontoxic photosensitizer (PS) with visible light to cause selective killing of microbial cells. The development of PACT to treat mycoses or kill fungi in the environment depends on identifying effective PS for the different pathogenic species and delivery systems able to expand and optimize their use. In the present study, the in vitro susceptibility of Cryptococcus neoformans melanized cells to the photodynamic effects of the PS agent ClAlPc in nanoemulsion (ClAlPc/NE) was examined. Cells were killed in a PS concentration- and light dose-dependent manner. Treatment with ClAlPc/NE, using PS concentrations (e.g. 4.5 μm) and light doses (e.g. 10 J cm(-2)) compatible with PACT, resulted in a reduction of up to 6 logs in survival. Washing the cells to remove unbound PS before light exposure did not inhibit fungal photodynamic inactivation. Internalization of ClAlPc by C. neoformans was confirmed by confocal fluorescence microscopy, and the degree of uptake was dependent on PS concentration.     

From biomass wastes to large-area, high-quality, N-doped graphene: catalyst-free carbonization of chitosan coatings on arbitrary substrates

Chitosan is a N-containing biopolymer that can form high-quality films on glass, quartz, metals and other hydrophilic surfaces. Pyrolysis of chitosan films under argon at 800 °C and under inert atmosphere gives rise to high-quality single layer N-doped graphene films (over 99% transmittance) as evidenced by XPS, Raman spectroscopy, and TEM imaging.     

Synergy between the metal nanoparticles and the support for the hydrogenation of functionalized carboxylic acids to diols on Ru/TiO2

Ruthenium nanoparticles supported on titania are over three times more active than conventional ruthenium on carbon for the hydrogenation of lactic acid. This superior catalytic activity can be due to a combined action of small ruthenium nanoparticles and the titania support.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.