Proximity Drawings of High-Degree Trees

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set

Is the New Resonance Spin 0 or 2? Taking a Step Forward in the Higgs Boson Discovery

The observation of a new boson of mass \sim 125\gev at the CERN LHC may finally have revealed the existence of a Higgs boson. Now we have the opportunity to scrutinize its properties, determining its quantum numbers and couplings to the standard model particles, in order to confirm or not its discovery. We show that by the end of the 8 TeV run, combining the entire data sets of ATLAS and CMS, it will be possible to discriminate between the following discovery alternatives: a scalar $J^P=0^+$ or a tensor $J^P=2^+$ particle with minimal couplings to photons, at a $5\sigma$ statistical confidence level at least, using only diphotons events. Our results are based on the calculation of a center-edge asymmetry measure of the reconstructed {\it sPlot} scattering polar angle of the diphotons. The results based on asymmetries are shown to be rather robust against systematic uncertainties with comparable discrimination power to a log likelihood ratio statistic.Comment: 11 pages, 6 figures, 1 table. References added, minor typos correcte

Transverse voltage in zero external magnetic fields, its scaling and violation of the time reversal symmetry in MgB2

The longitudinal and transverse voltages (resistances) have been measured for MgB$_2$ in zero external magnetic fields. Samples were prepared in the form of thin film and patterned into the usual Hall bar shape. In close vicinity of the critical temperature T$_c$ non-zero transverse resistance has been observed. Its dependence on the transport current has been also studied. New scaling between transverse and longitudinal resistivities has been observed in the form $\rho{_{xy}}\sim d\rho{_{xx}}/dT$. Several models for explanation of the observed transverse resistances and breaking of reciprocity theorem are discussed. One of the most promising explanation is based on the idea of time-reversal symmetry violation

Defect formation and local gauge invariance

We propose a new mechanism for formation of topological defects in a U(1) model with a local gauge symmetry. This mechanism leads to definite predictions, which are qualitatively different from those of the Kibble-Zurek mechanism of global theories. We confirm these predictions in numerical simulations, and they can also be tested in superconductor experiments. We believe that the mechanism generalizes to more complicated theories.Comment: REVTeX, 4 pages, 2 figures. The explicit form of the Hamiltonian and the equations of motion added. To appear in PRL (http://prl.aps.org/

Trapping in complex networks

We investigate the trapping problem in Erdos-Renyi (ER) and Scale-Free (SF) networks. We calculate the evolution of the particle density $\rho(t)$ of random walkers in the presence of one or multiple traps with concentration $c$. We show using theory and simulations that in ER networks, while for short times $\rho(t) \propto \exp(-Act)$, for longer times $\rho(t)$ exhibits a more complex behavior, with explicit dependence on both the number of traps and the size of the network. In SF networks we reveal the significant impact of the trap's location: $\rho(t)$ is drastically different when a trap is placed on a random node compared to the case of the trap being on the node with the maximum connectivity. For the latter case we find \rho(t)\propto\exp\left[-At/N^\frac{\gamma-2}{\gamma-1}\av{k}\right] for all $\gamma>2$, where $\gamma$ is the exponent of the degree distribution $P(k)\propto k^{-\gamma}$.Comment: Appendix adde

Transport in networks with multiple sources and sinks

We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, increasing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n* above which the transport is less efficient. For current, the typical decrease in the length of the connecting paths for large n compensates for the effect of the bottlenecks. We also derive an expression for the average flow as a function of n under the common limitation that transport takes place between specific pairs of sources and sinks

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem