765,063 research outputs found
On partitions avoiding 3-crossings
A partition on has a crossing if there exists
such that and are in the same block, and are in the
same block, but and are not in the same block. Recently, Chen et
al. refined this classical notion by introducing -crossings, for any integer
. In this new terminology, a classical crossing is a 2-crossing. The number
of partitions of avoiding 2-crossings is well-known to be the th
Catalan number . This raises the question of
counting -noncrossing partitions for . We prove that the sequence
counting 3-noncrossing partitions is P-recursive, that is, satisfies a linear
recurrence relation with polynomial coefficients. We give explicitly such a
recursion. However, we conjecture that -noncrossing partitions are not
P-recursive, for
Improved bounds for the crossing numbers of K_m,n and K_n
It has been long--conjectured that the crossing number cr(K_m,n) of the
complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):=
floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing
conjecture states that the crossing number cr(K_n) of the complete graph K_n
equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In
this paper we show the following improved bounds on the asymptotic ratios of
these crossing numbers and their conjectured values:
(i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1);
(ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and
(iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83.
The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8,
respectively. These improved bounds are obtained as a consequence of the new
bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for
cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to
set up a quadratic program on 6! variables whose minimum p satisfies
cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic
optimization techniques combined with a bit of invariant theory of permutation
groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure
Flip Distance to a Non-crossing Perfect Matching
A perfect straight-line matching on a finite set of points in the
plane is a set of segments such that each point in is an endpoint of
exactly one segment. is non-crossing if no two segments in cross each
other. Given a perfect straight-line matching with at least one crossing,
we can remove this crossing by a flip operation. The flip operation removes two
crossing segments on a point set and adds two non-crossing segments to
attain a new perfect matching . It is well known that after a finite number
of flips, a non-crossing matching is attained and no further flip is possible.
However, prior to this work, no non-trivial upper bound on the number of flips
was known. If (resp.~) is the maximum length of the longest
(resp.~shortest) sequence of flips starting from any matching of size , we
show that and (resp.~ and
)
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
New Observables for Parity Violation in Atoms: Energy Shifts in External Electric Fields
We consider hydrogen-like atoms in unstable levels of principal quantum
number n=2, confined to a finite size region in a non-homogeneous electric
field carrying handedness. The interplay between the internal degrees of
freedom of the atoms and the external ones of their c.m. motion can produce
P-odd contributions to the eigenenergies. The nominal order of such shifts is
10^-8 Hz. Typically such energy shifts depend linearly on the small P-violation
parameters delta_i similarequal 10^-12 (i=1,2), essentially the ratios of the
P-violating mixing matrix elements of the 2S and 2P states over the Lamb shift,
with i=1 (i=2) corresponding to the nuclear spin independent (dependent) term.
We show how such energy shifts can be enhanced by a factor of similarequal 10^6
in a resonance like way for special field configurations where a crossing of
unstable levels occurs, leading to P-violating effects proportional to
squareroot{delta_i}. Measurements of such effects can give information
concerning the ``spin crisis'' of the nucleons.Comment: 6 pages, 3 figures, LaTeX, submitted to Phys. Lett.
Particle alignments and shape change in Ge and Ge
The structure of the nuclei Ge and Ge is studied
by the shell model on a spherical basis. The calculations with an extended
Hamiltonian in the configuration space
(, , , ) succeed in reproducing
experimental energy levels, moments of inertia and moments in Ge isotopes.
Using the reliable wave functions, this paper investigates particle alignments
and nuclear shapes in Ge and Ge.
It is shown that structural changes in the four sequences of the positive-
and negative-parity yrast states with even and odd are caused by
various types of particle alignments in the orbit.
The nuclear shape is investigated by calculating spectroscopic moments of
the first and second states, and moreover the triaxiality is examined by
the constrained Hatree-Fock method.
The changes of the first band crossing and the nuclear deformation depending
on the neutron number are discussed.Comment: 18 pages, 21 figures; submitted to Phys. Rev.
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