11,835 research outputs found
Potential Vorticity Mixing in a Tangled Magnetic Field
A theory of potential vorticity (PV) mixing in a disordered (tangled)
magnetic field is presented. The analysis is in the context of -plane
MHD, with a special focus on the physics of momentum transport in the stably
stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution
by vorticity advection and tilting of magnetic fields is proposed. In the case
of weak-field perturbations, quasi-linear theory predicts that the Reynolds and
magnetic stresses balance as turbulence Alfv\'enizes for a larger mean magnetic
field. Jet formation is explored quantitatively in the mean field-resistivity
parameter space. However, since even a modest mean magnetic field leads to
large magnetic perturbations for large magnetic Reynolds number, the physically
relevant case is that of a strong but disordered field. We show that numerical
calculations indicate that the Reynolds stress is modified well before
Alfv\'enization -- i.e. before fluid and magnetic energies balance. To
understand these trends, a double-average model of PV mixing in a stochastic
magnetic field is developed. Calculations indicate that mean-square fields
strongly modify Reynolds stress phase coherence and also induce a magnetic drag
on zonal flows. The physics of transport reduction by tangled fields is
elucidated and linked to the related quench of turbulent resistivity. We
propose a physical picture of the system as a resisto-elastic medium threaded
by a tangled magnetic network. Applications of the theory to momentum transport
in the tachocline and other systems are discussed in detail.Comment: 17 pages, 10 figures, 2 table
Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
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