On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schr\"odinger equation

We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schr\"odinger equation. In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e. a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a self-similar manner

On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schr\"odinger equation

We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schr\"odinger equation. The solutions that we construct have finite mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions

Quantum recurrence of a subspace and operator-valued Schur functions

A notion of monitored recurrence for discrete-time quantum processes was recently introduced in [Commun. Math. Phys., DOI 10.1007/s00220-012-1645-2] (see also arXiv:1202.3903) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in the above mentioned paper. The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time. This notion of subspace recurrence also links the concept of expected return time to an Aharonov-Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counterintuitive behavior of quantum systems. All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.Comment: 40 pages, 8 figure

Ant foraging and minimal paths in simple graphs

Ants are known to be able to find paths of minimal length between the nest and food sources. The deposit of pheromones while they search for food and their chemotactical response to them has been proposed as a crucial element in the mechanism for finding minimal paths. We investigate both individual and collective behavior of ants in some simple networks representing basic mazes. The character of the graphs considered is such that it allows a fully rigorous mathematical treatment via analysis of some markovian processes in terms of which the evolution can be represented. Our analytical and computational results show that in order for the ants to follow shortest paths between nest and food, it is necessary to superimpose to the ants' random walk the chemotactic reinforcement. It is also needed a certain degree of persistence so that ants tend to move preferably without changing their direction much. It is also important the number of ants, since we will show that the speed for finding minimal paths increases very fast with it.Comment: 39 pages, 13 figure

The partition dimension of corona product graphs

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set of $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension $dim(G)$ of $G$ is the minimum cardinality of any resolving set of $G$. Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of vertices of a connected graph $G$, the partition representation of a vertex $v$ of $G$, with respect to the partition $\Pi$ is the vector $r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$, $1\leq i\leq t$, represents the distance between the vertex $v$ and the set $P_i$, that is $d(v,P_i)=\min_{u\in P_i}\{d(v,u)\}$. $\Pi$ is a resolving partition for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|\Pi)\ne r(v|\Pi)$. The partition dimension $pd(G)$ of $G$ is the minimum number of sets in any resolving partition for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$ respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and then joining by an edge, all the vertices from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. Here we study the relationship between $pd(G\odot H)$ and several parameters of the graphs $G\odot H$, $G$ and $H$, including $dim(G\odot H)$, $pd(G)$ and $pd(H)$

On the super domination number of graphs

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered

Computing the metric dimension of a graph from primary subgraphs

Let $G$ be a connected graph. Given an ordered set $W = \{w_1, w_2,\dots w_k\}\subseteq V(G)$ and a vertex $u\in V(G)$, the representation of $u$ with respect to $W$ is the ordered $k$-tuple $(d(u,w_1), d(u,w_2),\dots,$ $d(u,w_k))$, where $d(u,w_i)$ denotes the distance between $u$ and $w_i$. The set $W$ is a metric generator for $G$ if every two different vertices of $G$ have distinct representations. A minimum cardinality metric generator is called a \emph{metric basis} of $G$ and its cardinality is called the \emph{metric dimension} of G. It is well known that the problem of finding the metric dimension of a graph is NP-Hard. In this paper we obtain closed formulae for the metric dimension of graphs with cut vertices. The main results are applied to specific constructions including rooted product graphs, corona product graphs, block graphs and chains of graphs.Comment: 18 page

On the super domination number of lexicographic product graphs

The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$ we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard

The Simultaneous Strong Metric Dimension of Graph Families

Let ${\cal G}$ be a family of graphs defined on a common (labeled) vertex set $V$. A set $S\subset V$ is said to be a simultaneous strong metric generator for ${\cal G}$ if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for ${\cal G}$, denoted by $Sd_s({\cal G})$, is called the simultaneous strong metric dimension of ${\cal G}$. We obtain general results on $Sd_s({\cal G})$ for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is $NP$-hard, even when restricted to families of trees.Comment: arXiv admin note: text overlap with arXiv:1312.1987 by other author

The local metric dimension of the lexicographic product of graphs

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \circ \mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \circ \mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\mathcal{H}$