15,479 research outputs found
Cycles with consecutive odd lengths
It is proved that there exists an absolute constant c > 0 such that for every
natural number k, every non-bipartite 2-connected graph with average degree at
least ck contains k cycles with consecutive odd lengths. This implies the
existence of the absolute constant d > 0 that every non-bipartite 2-connected
graph with minimum degree at least dk contains cycles of all lengths modulo k,
thus providing an answer (in a strong form) to a question of Thomassen. Both
results are sharp up to the constant factors.Comment: 7 page
On -superharmonic functions and some geometric applications
In this paper we study asymptotic behavior of -superharmonic functions at
isolated singularity using the Wolff potential and -capacity estimates in
nonlinear potential theory. Our results are inspired by and extend those of
Arsove-Huber and Taliaferro in 2 dimensions. To study -superharmonic
functions we use a new notion of -thinness by -capacity motivated by a
type of Wiener criterion in Arsove-Huber's paper. To extend Taliaferro's work,
we employ the Adams-Moser-Trudinger inequality for the Wolff potential, which
is inspired by the one used by Brezis-Merle. For geometric applications, we
study the asymptotic end behavior of complete conformally flat manifolds as
well as complete properly embedded hypersurfaces in hyperbolic space. In both
geometric applications the strong -capacity lower bound estimate of Gehring
in 1961 is brilliantly used. These geometric applications seem to elevate the
importance of -Laplace equations and make a closer tie to the classic
analysis developed in conformal geometry in general dimensions.Comment: 46 page
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