2,150 research outputs found
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 of a connected graph , the
metric representation of a vertex of with respect to is the vector
, where ,
denotes the distance between and . is a resolving set of if
for every pair of vertices of , . The metric
dimension of is the minimum cardinality of any resolving set of
. Given an ordered partition of vertices of a
connected graph , the partition representation of a vertex of , with
respect to the partition is the vector
, where , ,
represents the distance between the vertex and the set , that is
. is a resolving partition for if
for every pair of vertices of , . The partition
dimension of is the minimum number of sets in any resolving
partition for . Let and be two graphs of order and
respectively. The corona product is defined as the graph obtained
from and by taking one copy of and copies of and then
joining by an edge, all the vertices from the -copy of with the
-vertex of . Here we study the relationship between
and several parameters of the graphs , and , including
, and
On the super domination number of graphs
The open neighbourhood of a vertex of a graph is the set
consisting of all vertices adjacent to in . For , we
define . A set is called a
super dominating set of if for every vertex , there
exists such that . The super domination
number of is the minimum cardinality among all super dominating sets in
. In this article, we obtain closed formulas and tight bounds for the super
domination number of in terms of several invariants of . 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 be a connected graph. Given an ordered set and a vertex , the representation of with
respect to is the ordered -tuple
, where denotes the distance between and . The
set is a metric generator for if every two different vertices of
have distinct representations. A minimum cardinality metric generator is called
a \emph{metric basis} of 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 of a graph is the set of all
vertices adjacent to in . For we define
. A set is called a super
dominating set if for every vertex , there exists
such that . The super domination number of is
the minimum cardinality among all super dominating sets in . 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 be a family of graphs defined on a common (labeled) vertex set
. A set is said to be a simultaneous strong metric generator
for if it is a strong metric generator for every graph of the
family. The minimum cardinality among all simultaneous strong metric generators
for , denoted by , is called the simultaneous strong
metric dimension of . We obtain general results on
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 -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 of a connected
graph of order and a family composed by graphs. We
show that the local metric dimension of can be expressed
in terms of the true twin equivalence classes of and the local adjacency
dimension of the graphs in
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