37,230 research outputs found
On colored set partitions of type
Generalizing Reiner's notion of set partitions of type , we define
colored -partitions by coloring the elements in and not in the zero-block
respectively. Considering the generating function of colored -partitions,
we get the exact formulas for the expectation and variance of the number of
non-zero-blocks in a random colored -partition. We find an asymptotic
expression of the total number of colored -partitions up to an error of
, and prove that the centralized and normalized
number of non-zero-blocks is asymptotic normal over colored -partitions.Comment: 10 page
The Real-Rootedness and Log-concavities of Coordinator Polynomials of Weyl Group Lattices
It is well-known that the coordinator polynomials of the classical root
lattice of type and those of type are real-rooted. They can be
obtained, either by the Aissen-Schoenberg-Whitney theorem, or from their
recurrence relations. In this paper, we develop a trigonometric substitution
approach which can be used to establish the real-rootedness of coordinator
polynomials of type . We also find the coordinator polynomials of type
are not real-rooted in general. As a conclusion, we obtain that all
coordinator polynomials of Weyl group lattices are log-concave.Comment: 8 page
Root geometry of polynomial sequences III: Type with positive coefficients
In this paper, we study the root distribution of some univariate polynomials
satisfying a recurrence of order two with linear polynomial
coefficients over positive numbers. We discover a sufficient and necessary
condition for the overall real-rootedness of all the polynomials, in terms of
the polynomial coefficients of the recurrence. Moreover, in the real-rooted
case, we find the set of limits of zeros, which turns out to be the union of a
closed interval and one or two isolated points; when non-real-rooted polynomial
exists, we present a sufficient condition under which every polynomial with
large has a real zero.Comment: 14page, 3 figure
Surface embedding of non-bipartite -extendable graphs
We find the minimum number for any surface , such
that every -embeddable non-bipartite graph is not -extendable. In
particular, we construct the so-called bow-tie graphs , and
show that they are -extendable. This confirms the existence of an infinite
number of -extendable non-bipartite graphs which can be embedded in the
Klein bottle.Comment: 17 pages, 5 figure
A Note on -Factors of Regular Graphs
Let be an odd integer, and an even integer. In this note, we present
-regular graphs which have no -factors for all . This gives a negative answer to a problem posed by Akbari and
Kano recently
The maximum number of perfect matchings of semi-regular graphs
Let be an even integer, and . In this
paper, we prove that every -graph of order contains
disjoint perfect matchings. This result is sharp in the
sense that (i) there exists a -graph containing exactly
disjoint perfect matchings, and that (ii) there exists a
-graph without perfect matchings for each . As a
consequence, for any integer , every -graph of order
contains disjoint perfect matchings. This extends Csaba
et~al.'s breathe-taking result that every -regular graph of sufficiently
large order is -factorizable, generalizes Zhang and Zhu's result that every
-regular graph of order contains disjoint perfect
matchings, and improves Hou's result that for all , every
-graph of order contains
disjoint perfect matchings.Comment: 30 pages, 9 figure
Piecewise interlacing zeros of polynomials
We introduce the concept of piecewise interlacing zeros for studying the
relation of root distribution of two polynomials. The concept is pregnant with
an idea of confirming the real-rootedness of polynomials in a sequence. Roughly
speaking, one constructs a collection of disjoint intervals such that one may
show by induction that consecutive polynomials have interlacing zeros over each
of the intervals. We confirm the real-rootedness of some polynomials satisfying
a recurrence with linear polynomial coefficients. This extends Gross et al.'s
work where one of the polynomial coefficients is a constant.Comment: 18 pages, 6 figure
The Tutte's condition in terms of graph factors
Let be a connected general graph of even order, with a function . We obtain that satisfies the Tutte's condition with
respect to if and only if contains an -factor for any function
such that for each , where the set consists of the integer and all positive
odd integers less than , and the set consists of positive odd
integers less than or equal to . We also obtain a characterization for
graphs of odd order satisfying the Tutte's condition with respect to a
function.Comment: 5 page
A Debris Disk Around An Isolated Young Neutron Star
Pulsars are rotating, magnetized neutron stars that are born in supernova
explosions following the collapse of the cores of massive stars. If some of the
explosion ejecta fails to escape, it may fall back onto the neutron star or it
may possess sufficient angular momentum to form a disk. Such 'fallback' is both
a general prediction of current supernova models and, if the material pushes
the neutron star over its stability limit, a possible mode of black hole
formation. Fallback disks could dramatically affect the early evolution of
pulsars, yet there are few observational constraints on whether significant
fallback occurs or even the actual existence of such disks. Here we report the
discovery of mid-infrared emission from a cool disk around an isolated young
X-ray pulsar. The disk does not power the pulsar's X-ray emission but is
passively illuminated by these X-rays. The estimated mass of the disk is of
order 10 Earth masses, and its lifetime (at least a million years)
significantly exceeds the spin-down age of the pulsar, supporting a supernova
fallback origin. The disk resembles protoplanetary disks seen around ordinary
young stars, suggesting the possibility of planet formation around young
neutron stars.Comment: 5 pages, 3 figures. To appear in Nature (6 Apr 2006
Degenerate Motions in Multicamera Cluster SLAM with Non-overlapping Fields of View
An analysis of the relative motion and point feature model configurations
leading to solution degeneracy is presented, for the case of a Simultaneous
Localization and Mapping system using multicamera clusters with non-overlapping
fields-of-view. The SLAM optimization system seeks to minimize image space
reprojection error and is formulated for a cluster containing any number of
component cameras, observing any number of point features over two keyframes.
The measurement Jacobian is transformed to expose a reduced-dimension
representation such that the degeneracy of the system can be determined by the
rank of a dense submatrix. A set of relative motions sufficient for degeneracy
are identified for certain cluster configurations, independent of target model
geometry. Furthermore, it is shown that increasing the number of cameras within
the cluster and observing features across different cameras over the two
keyframes reduces the size of the degenerate motion sets significantly.Comment: 18 pages, 18 figure
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