Edge partitions of optimal 2-plane and 3-plane graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests.     

The structure of the fractional powers of the noncommutative Fourier law

In the recent years, there has been a lot of interest in fractional diffusion and fractional evolution problems. The spectral theory on the S‐spectrum turned out to be an important tool to define new fractional diffusion operators stating from the Fourier law for nonhomogeneous materials. Precisely, let e_?, e_?=1,2,3 be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation is obtained replacing the Fourier law given by into the conservation of energy law. In this paper, we investigate the structure of the fractional powers of the vector operator T, with homogeneous Dirichlet boundary conditions. Recently, we have found sufficient conditions on the coefficients a, b, such that the fractional powers of T exist in the sense of the S‐spectrum approach. In this paper, we show that under a different set of conditions on the coefficients a, b, c, the fractional powers of T have a different structure.     

Interaction between Spirosilanes and Lewis Bases: from Coordination to Frustration

In this work, the interaction between Lewis bases, especially N-heterocyclic carbenes (NHCs), and hindered neutral silicon derivatives featuring high Lewis acidity is described. The formation of normal and abnormal Lewis adducts could be controlled by varying the acidity of the corresponding tetravalent spiro organosilane. Some DFT calculations permitted to gain insight into the thermodynamics of the NHC–spirosilane interaction featuring various NHCs differing in size and σ-donor capacity. Spirosilanes are introduced as new Lewis partners in frustrated Lewis pair (FLP) chemistry and some FLP-type reactivities are presented, in particular the activation of formaldehyde that could occur with both hindered NHCs and phosphines.     

Topical Issue on Thermal Non-Equilibrium Phenomena in Soft Matter

The European Physical Journal E - The transmission and the reflection of a shock wave by one or several impurities in viscoelastic bead chains is studied in the present paper. There is only one...     

Magnetic Properties of Two New Fe4 Single‐Molecule Magnets in the Solid State and in Frozen Solution

Two novel tetranuclear, star‐shaped iron(III) clusters, [Fe₄(acac)₆(Br‐mp)₂] and [Fe^III₄(acac)₆(tmp)₂], are described. Both have S=5 ground states resulting from antiferromagnetic nearest‐neighbour superexchange interactions, with J=?8.2 cm^?1 and J=?8.5 cm^?1 for 1 and 2 , respectively. Energy barriers for the relaxation of the magnetisation of approximately 12 cm^?1 were derived from AC susceptibility measurements. Magnetic resonance measurements revealed a zero‐field splitting parameter D=?0.34 cm^?1 for both complexes. AC susceptibility measurements in solution demonstrated that the complexes are reasonably stable in solution. Interestingly, the magnetisation relaxation slows down significantly in frozen solution, in contrast to what is generally observed for single‐molecule magnets. This was shown to result from a large increase in τ₀, the prefactor in the Arrhenius equation, with the energy barrier remaining unchanged.     

Stability and reactivity of ϵ-χ-θ iron carbide catalyst phases in Fischer-Tropsch synthesis: controlling μ(C)

The stability and reactivity of ?, χ, and θ iron carbide phases in Fischer-Tropsch synthesis (FTS) catalysts as a function of relevant reaction conditions was investigated by a synergistic combination of experimental and theoretical methods. Combined in situ X-ray Absorption Fine Structure Spectroscopy/X-ray Diffraction/Raman Spectroscopy was applied to study Fe-based catalysts during pretreatment and, for the first time, at relevant high pressure Fischer-Tropsch synthesis conditions, while Density Functional Theory calculations formed a fundamental basis for understanding the influence of pretreatment and FTS conditions on the formation of bulk iron carbide phases. By combining theory and experiment, it was found that the formation of θ-Fe(3)C, χ-Fe(5)C(2), and ?-carbides can be explained by their relative thermodynamic stability as imposed by gas phase composition and temperature. Furthermore, it was shown that a significant part of the Fe phases was present as amorphous carbide phases during high pressure FTS, sometimes in an equivalent amount to the crystalline iron carbide fraction. A catalyst containing mainly crystalline χ-Fe(5)C(2) was highly susceptible to oxidation during FTS conditions, while a catalyst containing θ-Fe(3)C and amorphous carbide phases showed a lower activity and selectivity, mainly due to the buildup of carbonaceous deposits on the catalyst surface, suggesting that amorphous phases and the resulting textural properties play an important role in determining final catalyst performance. The findings further uncovered the thermodynamic and kinetic factors inducing the ?-χ-θ carbide transformation as a function of the carbon chemical potential μ(C).     

1-Bromo-2-(diphenylphosphinoyl)ethyne and 1-bromo-2-(p-tolylsulfinyl)ethyne: versatile reagents eventually leading to benzocyclotrimers

The title compounds form Diels-Alder cycloadducts with a number of dienes, which can be transformed into vic-bromo(trimethylstannyl)olefins, ultimate precursors for the synthesis of benzocyclotrimers via copper-mediated cyclotrimerization. The overall result is a formal condensation of three diatomic carbons with three dienes.     

Multimodal imaging: an evaluation of univariate and multivariate methods for simultaneous EEG/fMRI

The combination of electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) has been proposed as a tool to study brain dynamics with both high temporal and high spatial resolution. Multimodal imaging techniques rely on the assumption of a common neuronal source for the different recorded signals. In order to maximally exploit the combination of these techniques, one needs to understand the coupling (i.e., the relation) between electroencephalographic (EEG) and fMRI blood oxygen level-dependent (BOLD) signals.