No abstract

We consider the transformation reversing all arcs of a subset $X$ of the vertex set of a tournament $T$. The \emph{index} of $T$, denoted by $i(T)$, is the smallest number of subsets that must be reversed to make $T$ acyclic. It turns out that critical tournaments and $(-1)$-critical tournaments can be defined in terms of inversions (at most two for the former, at most four for the latter). We interpret $i(T)$ as the minimum distance of $T$ to the transitive tournaments on the same vertex set, and we interpret the distance between two tournaments $T$ and $T'$ as the \emph{Boolean dimension} of a graph, namely the Boolean sum of $T$ and $T'$. On $n$ vertices, the maximum distance is at most $n-1$, whereas $i(n)$, the maximum of $i(T)$ over the tournaments on $n$ vertices, satisfies $\frac {n-1}{2} - \log_{2}n \leq i(n) \leq n-3$, for $n \geq 4$. Let $ \mathcal{I}_{m}^{< \omega}$ (resp. $\mathcal{I}_{m}^{\leq \omega}$) be the class of finite (resp. at most countable) tournaments $T$ such that $i(T) \leq m$. The class $\mathcal {I}_{m}^{< \omega}$ is determined by finitely many obstructions. We give a morphological description of the members of $\mathcal {I}_{1}^{< \omega}$ and a description of the critical obstructions. We give an explicit description of an universal tournament of the class $\mathcal{I}_{m}^{\leq \omega}$.Comment: 6 page

Abstract. This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting posetΩ. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members ofΩ are the −2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members ofΩ.

Recently, Herra´ez et al. proposed a new correlation equation, which introduces a correcting polynomial as an exponential-acting upon the molar fraction of one of mixture components. This equation is found to be widely applicable with more reasonable accuracy in systems exhibiting monotonous distribution than those presenting an extremum. In previous works, we have found that the first adjustable parameter of this equation is a universal exponent (0.5 or 1) for dioxane-water and isobutyric acid-water mixtures characterising the presence of solute-solute or solute-solvent interaction at very high dilution. In this work, we have tested this equation in 48 systems, and we have noted that several data on binary mixtures exhibiting monotonous variation are well represented by the two sides of this equation. We have mathematically justified the four distinct values of the universal exponent classifying the nature of each solute-solvent couple in dilute region.

Excess properties calculated from the experimental values of densities and viscosities have been presented in the previous work. These experimental values can also lead us to test different correlation equations as well as their corresponding relative functions. Inspection of the Arrhenius activation energy Ea and the enthalpy of activation of viscous flow ΔH* found very close values, here we can define partial molar activation energy Ea 1 and Ea 2 for N,N-dimethylacetamide and methanol respectively along with their individual contribution separately. Correlation between the two Arrhenius parameters of viscosity in all compositions shows existence of distinct behaviours separated by particular mole fractions of N,N-dimethylacetamide. In addition, the correlation between Arrhenius parameters reveals interesting Arrhenius temperature which is closely related to the vaporisation temperature in the liquid vapour equilibrium and the limiting corresponding partial molar properties that can permit us to estimate the boiling points of the pure components.