On global location-domination in graphs

A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the location-domination number $\lambda(G)$. An LD-set $S$ of a graph $G$ is global if it is an LD-set of both $G$ and its complement $\overline{G}$. The global location-domination number $\lambda_g(G)$ is the minimum cardinality of a global LD-set of $G$. In this work, we give some relations between locating-dominating sets and the location-domination number in a graph and its complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference

Confinement-induced resonances for a two-component ultracold atom gas in arbitrary quasi-one-dimensional traps

We solve the two-particle s-wave scattering problem for ultracold atom gases confined in arbitrary quasi-one-dimensional trapping potentials, allowing for two different atom species. As a consequence, the center-of-mass and relative degrees of freedom do not factorize. We derive bound-state solutions and obtain the general scattering solution, which exhibits several resonances in the 1D scattering length induced by the confinement. We apply our formalism to two experimentally relevant cases: (i) interspecies scattering in a two-species mixture, and (ii) the two-body problem for a single species in a non-parabolic trap.Comment: 22 pages, 3 figure

Extremal Graph Theory for Metric Dimension and Diameter

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\mathcal{G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in $\mathcal{G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in $\mathcal{G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in $\mathcal{G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known

Universal resources for approximate and stochastic measurement-based quantum computation

We investigate which quantum states can serve as universal resources for approximate and stochastic measurement-based quantum computation, in the sense that any quantum state can be generated from a given resource by means of single-qubit (local) operations assisted by classical communication. More precisely, we consider the approximate and stochastic generation of states, resulting e.g. from a restriction to finite measurement settings or from possible imperfections in the resources or local operations. We show that entanglement-based criteria for universality obtained for the exact, deterministic case can be lifted to the much more general approximate, stochastic case, moving from the idealized situation considered in previous works, to the practically relevant context of non-perfect state preparation. We find that any entanglement measure fulfilling some basic requirements needs to reach its maximum value on some element of an approximate, stochastic universal family of resource states, as the resource size grows. This allows us to rule out various families of states as being approximate, stochastic universal. We provide examples of resources that are efficient approximate universal, but not exact deterministic universal. We also study the robustness of universal resources for measurement-based quantum computation under realistic assumptions about the (imperfect) generation and manipulation of entangled states, giving an explicit expression for the impact that errors made in the preparation of the resource have on the possibility to use it for universal approximate and stochastic state preparation. Finally, we discuss the relation between our entanglement-based criteria and recent results regarding the uselessness of states with a high degree of geometric entanglement as universal resources.Comment: 17 pages; abstract shortened with respect to the published version to respect the arXiv limit of 1,920 character

Necking in 2D incompressible polyconvex materials: theoretical framework and numerical simulations

We show examples of 2D incompressible isotropic homogeneous hyperelastic materials with a poly-convex stored-energy function that present necking. The construction of the stored-energy function of amaterial satisfying all those properties requires a fine search. We used the software Algencan to perform numerical experiments and visualize necking for the examples constructed. The algorithm is based on minimization of the elastic energy under the nonconvex constraint of incompressibility