On Domination Number and Distance in Graphs

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over all dominating sets of $G$. It is known that $\gamma(G) \ge \frac{1}{3}(diam(G)+1)$, where $diam(G)$ denotes the diameter of $G$. Define $C_r$ as the largest constant such that $\gamma(G) \ge C_r \sum_{1 \le i < j \le r}d(x_i, x_j)$ for any $r$ vertices of an arbitrary connected graph $G$; then $C_2=\frac{1}{3}$ in this view. The main result of this paper is that $C_r=\frac{1}{r(r-1)}$ for $r\geq 3$. It immediately follows that $\gamma(G)\geq \mu(G)=\frac{1}{n(n-1)}W(G)$, where $\mu(G)$ and $W(G)$ are respectively the average distance and the Wiener index of $G$ of order $n$. As an application of our main result, we prove a conjecture of DeLaVi\~{n}a et al.\;that $\gamma(G)\geq \frac{1}{2}(ecc_G(B)+1)$, where $ecc_G(B)$ denotes the eccentricity of the boundary of an arbitrary connected graph $G$.Comment: 5 pages, 2 figure

Partial Observability and its Consistency for PDEs

In this paper, a quantitative measure of partial observability is defined for PDEs. The quantity is proved to be consistent if the PDE is approximated using well-posed approximation schemes. A first order approximation of an unobservability index using an empirical Gramian is introduced. Several examples are presented to illustrate the concept of partial observability, including Burgers' equation and a one-dimensional nonlinear shallow water equation.Comment: 5 figures, 25 pages. arXiv admin note: substantial text overlap with arXiv:1111.584

The electromagnetic form factors of the proton in the timelike region

The reactions ppbar -> e+e- and e+e- -> ppbar are analyzed in the near-threshold region. Specific emphasis is put on the role played by the interaction in the initial- or final antinucleon-nucleon state which is taken into account rigorously. For that purpose a recently published NNbar potential derived within chiral effective field theory and fitted to results of a new partial-wave analysis of ppbar scattering data is employed. Our results provide strong support for the conjecture that the pronounced energy dependence of the e+e- ppbar cross section, seen in pertinent experiments, is primarily due to the ppbar interaction. Predictions for the proton electromagnetic form factors G_E and G_M in the timelike region, close to the NNbar threshold, and for spin-dependent observables are presented. The steep rise of the effective form factor for energies close to the ppbar threshold is explained solely in terms of the ppbar interaction. The corresponding experimental information is quantitatively described by our calculation.Comment: 14 pages, 11 figure

Strategic Asset Seeking and Innovation Performance: The Role of Innovation Capabilities and Host Country Institutions

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