11,052 research outputs found
k-Tuple_Total_Domination_in_Inflated_Graphs
The inflated graph of a graph with vertices is obtained
from by replacing every vertex of degree of by a clique, which is
isomorph to the complete graph , and each edge of is
replaced by an edge in such a way that , , and
two different edges of are replaced by non-adjacent edges of . For
integer , the -tuple total domination number of is the minimum cardinality of a -tuple total dominating set
of , which is a set of vertices in such that every vertex of is
adjacent to at least vertices in it. For existing this number, must the
minimum degree of is at least . Here, we study the -tuple total
domination number in inflated graphs when . First we prove that
, and then we
characterize graphs that the -tuple total domination number number of
is or . Then we find bounds for this number in the
inflated graph , when has a cut-edge or cut-vertex , in terms
on the -tuple total domination number of the inflated graphs of the
components of or -components of , respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs
A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation
In this paper we improve traditional steepest descent methods for the direct
minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels.
We first define a new inner product to equip the Sobolev space and derive
the corresponding gradient. Secondly, for the treatment of the mass
conservation constraint, we use a projection method that avoids more
complicated approaches based on modified energy functionals or traditional
normalization methods. The descent method with these two new ingredients is
studied theoretically in a Hilbert space setting and we give a proof of the
global existence and convergence in the asymptotic limit to a minimizer of the
GP energy. The new method is implemented in both finite difference and finite
element two-dimensional settings and used to compute various complex
configurations with vortices of rotating Bose-Einstein condensates. The new
Sobolev gradient method shows better numerical performances compared to
classical or gradient methods, especially when high rotation rates
are considered.Comment: to appear in SIAM J Sci Computin
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