On the Randi\'{c} index and conditional parameters of a graph

The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right.$$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243

ChPT parameters from tau-decay data

Using the updated ALEPH V-A spectral function from tau decays, we determine the lowest spectral moments of the left-right correlator and extract dynamical information on order parameters of the QCD chiral symmetry breaking. Uncertainties associated with violations of quark-hadron duality are estimated from the data, imposing all known short-distance constraints on a resonance-based parametrization. Employing proper pinched weight functions, we obtain an accurate determination of the effective chiral couplings L10 and C87 and the dimension-six and -eight contributions in the Operator Product Expansion.Comment: 5 pages, 3 figures, QCD2015 Montpellie

Polydispersity Effects in the Dynamics and Stability of Bubbling Flows

The occurrence of swarms of small bubbles in a variety of industrial systems enhances their performance. However, the effects that size polydispersity may produce on the stability of kinematic waves, the gain factor, mean bubble velocity, kinematic and dynamic wave velocities is, to our knowledge, not yet well established. We found that size polydispersity enhances the stability of a bubble column by a factor of about 23% as a function of frequency and for a particular type of bubble column. In this way our model predicts effects that might be verified experimentally but this, however, remain to be assessed. Our results reinforce the point of view advocated in this work in the sense that a description of a bubble column based on the concept of randomness of a bubble cloud and average properties of the fluid motion, may be a useful approach that has not been exploited in engineering systems.Comment: 11 pages, 2 figures, presented at the 3rd NEXT-SigmaPhi International Conference, 13-18 August, 2005, Kolymbari, Cret