10,069 research outputs found
The minimum distance of sets of points and the minimum socle degree
Let be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a
hyperplane. Let be the maximum number of points of
contained in any hyperplane, and let . If
is the ideal of , then in \cite{t1}
it is shown that for , has a lower bound expressed in terms
of some shift in the graded minimal free resolution of . In these notes we
show that this behavior is true in general, for any : , where and is the last module in
the graded minimal free resolution of . In the end we also prove that this
bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).Comment: 11 page
A commutative algebraic approach to the fitting problem
Given a finite set of points in not all contained
in a hyperplane, the "fitting problem" asks what is the maximum number
of these points that can fit in some hyperplane and what is (are)
the equation(s) of such hyperplane(s). If has the property that any
of its points span a hyperplane, then , where
is the index of nilpotency of an ideal constructed from the
homogeneous coordinates of the points of . Note that in
any two points span a line, and we find that the maximum number of collinear
points of any given set of points equals the index
of nilpotency of the corresponding ideal, plus one.Comment: 8 page
Overview of Neutron-Proton Pairing
The role of neutron-proton pairing correlations on the structure of nuclei
along the line is reviewed. Particular emphasis is placed on the
competition between isovector () and isoscalar ) pair fields. The
expected properties of these systems, in terms of pairing collective motion,
are assessed by different theoretical frameworks including schematic models,
realistic Shell Model and mean field approaches. The results are contrasted
with experimental data with the goal of establishing clear signals for the
existence of neutron-proton () condensates. We will show that there is
clear evidence for an isovector condensate as expected from isospin
invariance. However, and contrary to early expectations, a condensate of
deuteron-like pairs appears quite elusive and pairing collectivity in the
channel may only show in the form of a phonon. Arguments are presented for the
use of direct reactions, adding or removing an pair, as the most promising
tool to provide a definite answer to this intriguing question.Comment: 89 pages, 59 figures. Accepted for publication in Progress in
Particle and Nuclear Physics (ELSEVIER
Bounding invariants of fat points using a coding theory construction
Let Z \subseteq \proj{n} be a fat points scheme, and let be the
minimum distance of the linear code constructed from . We show that
imposes constraints (i.e., upper bounds) on some specific shifts in the graded
minimal free resolution of , the defining ideal of . We investigate
this relation in the case that the support of is a complete intersection;
when is reduced and a complete intersection we give lower bounds for
that improve upon known bounds.Comment: 18 pages, 1 figure; accepted in J. Pure Appl. Algebr
- β¦