149,603 research outputs found
Radon Numbers for Trees
Many interesting problems are obtained by attempting to generalize classical
results on convexity in Euclidean spaces to other convexity spaces, in
particular to convexity spaces on graphs. In this paper we consider
-convexity on graphs. A set of vertices in a graph is -convex
if every vertex not in has at most one neighbour in . More specifically,
we consider Radon numbers for -convexity in trees.
Tverberg's theorem states that every set of points in
can be partitioned into sets with intersecting convex hulls.
As a special case of Eckhoff's conjecture, we show that a similar result holds
for -convexity in trees.
A set of vertices in a graph is called free, if no vertex of has
more than one neighbour in . We prove an inequality relating the Radon
number for -convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure
The -log-convexity of Domb's polynomials
In this paper, we prove the -log-convexity of Domb's polynomials, which
was conjectured by Sun in the study of Ramanujan-Sato type series for powers of
. As a result, we obtain the log-convexity of Domb's numbers. Our proof is
based on the -log-convexity of Narayana polynomials of type and a
criterion for determining -log-convexity of self-reciprocal polynomials.Comment: arXiv admin note: substantial text overlap with arXiv:1308.273
Convexity preserving jump-diffusion models for option pricing
We investigate which jump-diffusion models are convexity preserving. The
study of convexity preserving models is motivated by monotonicity results for
such models in the volatility and in the jump parameters. We give a necessary
condition for convexity to be preserved in several-dimensional jump-diffusion
models. This necessary condition is then used to show that, within a large
class of possible models, the only convexity preserving models are the ones
with linear coefficients.Comment: 14 page
Convexity in a masure
Masures are generalizations of Bruhat-Tits buildings. They were introduced to
study Kac-Moody groups over ultrametric fields, which generalize reductive
groups over the same fields. If A and A are two apartments in a building, their
intersection is convex (as a subset of the finite dimensional affine space A)
and there exists an isomorphism from A to A fixing this intersection. We study
this question for masures and prove that the analogous statement is true in
some particular cases. We deduce a new axiomatic of masures, simpler than the
one given by Rousseau
Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems
Necessary and sufficient conditions for convexity and strong convexity,
respectively, of sublevel sets that are defined by finitely many real-valued
-maps are presented. A novel characterization of strongly convex sets
in terms of the so-called local quadratic support is proved. The results
concerning strong convexity are used to derive sufficient conditions for
attainable sets of continuous-time nonlinear systems to be strongly convex. An
application of these conditions is a novel method to over-approximate
attainable sets when strong convexity is present.Comment: 20 pages, 3 figure
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