4,210 research outputs found
Bounding sequence extremal functions with formations
An -formation is a concatenation of permutations of letters.
If is a sequence with distinct letters, then let be
the maximum length of any -sparse sequence with distinct letters which
has no subsequence isomorphic to . For every sequence define
, the formation width of , to be the minimum for which
there exists such that there is a subsequence isomorphic to in every
-formation. We use to prove upper bounds on
for sequences such that contains an alternation
with the same formation width as .
We generalize Nivasch's bounds on by showing that
and for every and , such that denotes the inverse Ackermann function.
Upper bounds on have been used in other
papers to bound the maximum number of edges in -quasiplanar graphs on
vertices with no pair of edges intersecting in more than points.
If is any sequence of the form such that is a letter,
is a nonempty sequence excluding with no repeated letters and is
obtained from by only moving the first letter of to another place in
, then we show that and . Furthermore we prove that
and for every .Comment: 25 page
Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field
In this paper we consider the minimum time population transfer problem for
the -component of the spin of a (spin 1/2) particle driven by a magnetic
field, controlled along the x axis, with bounded amplitude. On the Bloch sphere
(i.e. after a suitable Hopf projection), this problem can be attacked with
techniques of optimal syntheses on 2-D manifolds. Let be the two
energy levels, and the bound on the field amplitude. For
each couple of values and , we determine the time optimal synthesis
starting from the level and we provide the explicit expression of the time
optimal trajectories steering the state one to the state two, in terms of a
parameter that can be computed solving numerically a suitable equation. For
, every time optimal trajectory is bang-bang and in particular the
corresponding control is periodic with frequency of the order of the resonance
frequency . On the other side, for , the time optimal
trajectory steering the state one to the state two is bang-bang with exactly
one switching. Fixed we also prove that for the time needed to
reach the state two tends to zero. In the case there are time optimal
trajectories containing a singular arc. Finally we compare these results with
some known results of Khaneja, Brockett and Glaser and with those obtained by
controlling the magnetic field both on the and directions (or with one
external field, but in the rotating wave approximation). As byproduct we prove
that the qualitative shape of the time optimal synthesis presents different
patterns, that cyclically alternate as , giving a partial proof of a
conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte
Forbidden subposet problems for traces of set families
In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets form a
copy of a poset , if there exists a bijection such that for any the relation implies
. A family of sets is \textit{-free} if
it does not contain any copy of . The trace of a family on a
set is .
We introduce the following notions: is
-trace -free if for any -subset , the family
is -free and is trace -free if it is
-trace -free for all . As the first instances of these problems
we determine the maximum size of trace -free families, where is the
butterfly poset on four elements with and determine the
asymptotics of the maximum size of -trace -free families for
. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems
- …