16,898 research outputs found
Staircase diagrams and enumeration of smooth Schubert varieties
We enumerate smooth and rationally smooth Schubert varieties in the classical
finite types A, B, C, and D, extending Haiman's enumeration for type A. To do
this enumeration, we introduce a notion of staircase diagrams on a graph. These
combinatorial structures are collections of steps of irregular size, forming
interconnected staircases over the given graph. Over a Dynkin-Coxeter graph,
the set of "nearly-maximally labelled" staircase diagrams is in bijection with
the set of Schubert varieties with a complete Billey-Postnikov (BP)
decomposition. We can then use an earlier result of the authors showing that
all finite-type rationally smooth Schubert varieties have a complete BP
decomposition to finish the enumeration.Comment: 42 pages, 3 table
Treewidth of grid subsets
Let Q_n be the graph of n times n times n cube with all non-decreasing
diagonals (including the facial ones) in its constituent unit cubes. Suppose
that a subset S of V(Q_n) separates the left side of the cube from the right
side. We show that S induces a subgraph of tree-width at least n/sqrt{18}-1. We
use a generalization of this claim to prove that the vertex set of Q_n cannot
be partitioned to two parts, each of them inducing a subgraph of bounded
tree-width.Comment: 15 pages, no figure
Irreducible components of the equivariant punctual Hilbert schemes
Let H_{ab} be the equivariant Hilbert scheme parametrizing the 0-dimensional
subschemes of the affine plane invariant under the natural action of the
one-dimensional torus T_{ab}:={(t^{-b},t^a), t\in k^*}. We compute the
irreducible components of H_{ab}: they are in one-one correspondence with a set
of Hilbert functions. As a by-product of the proof, we give new proofs of
results by Ellingsrud and Stromme, namely the main lemma of the computation of
the Betti numbers of the Hilbert scheme H^l parametrizing the 0-dimensional
subschemes of the affine plane of length l and a description of
Bialynicki-Birula cells on H^l by means of explicit flat families. In
particular, we precise conditions of applications of this last description.Comment: 13 page
The Flip Diameter of Rectangulations and Convex Subdivisions
We study the configuration space of rectangulations and convex subdivisions
of points in the plane. It is shown that a sequence of
elementary flip and rotate operations can transform any rectangulation to any
other rectangulation on the same set of points. This bound is the best
possible for some point sets, while operations are sufficient and
necessary for others. Some of our bounds generalize to convex subdivisions of
points in the plane.Comment: 17 pages, 12 figures, an extended abstract has been presented at
LATIN 201
Periodic negative differential conductance in a single metallic nano-cage
We report a bi-polar multiple periodic negative differential conductance
(NDC) effect on a single cage-shaped Ru nanoparticle measured using scanning
tunneling spectroscopy. This phenomenon is assigned to the unique
multiply-connected cage architecture providing two (or more) defined routes for
charge flow through the cage. This, in turn, promotes a self- gating effect,
where electron charging of one route affects charge transport along a
neighboring channel, yielding a series of periodic NDC peaks. This picture is
established and analyzed here by a theoretical model
On the Oß-hull of a planar point set
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the Oß-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle ß. Given a set P of n points in the plane, we show how to maintain the Oß-hull of P while ß runs from 0 to p in T(n log n) time and O(n) space. With the same complexity, we also find the values of ß that maximize the area and the perimeter of the Oß-hull and, furthermore, we find the value of ß achieving the best fitting of the point set P with a two-joint chain of alternate interior angle ß.Peer ReviewedPostprint (author's final draft
Using secondary Upsilon invariants to rule out stable equivalence of knot complexes
Two Heegaard Floer knot complexes are called stably equivalent if an acyclic
complex can be added to each complex to make them filtered chain homotopy
equivalent. Hom showed that if two knots are concordant, then their knot
complexes are stably equivalent. Invariants of stable equivalence include the
concordance invariants , , and . Feller and
Krcatovich gave a relationship between the Upsilon invariants of torus knots.
We use secondary Upsilon invariants defined by Kim and Livingston to show that
these relations do not extend to stable equivalence.Comment: 16 pages, 7 figure
On the -hull of a planar point set
We study the -hull of a planar point set, a generalization of the
Orthogonal Convex Hull where the coordinate axes form an angle . Given a
set of points in the plane, we show how to maintain the -hull
of while runs from to in time and
space. With the same complexity, we also find the values of that
maximize the area and the perimeter of the -hull and, furthermore, we
find the value of achieving the best fitting of the point set with
a two-joint chain of alternate interior angle
Minimum Manhattan network problem in normed planes with polygonal balls: a factor 2.5 approximation algorithm
Let B be a centrally symmetric convex polygon of R^2 and || p - q || be the
distance between two points p,q in R^2 in the normed plane whose unit ball is
B. For a set T of n points (terminals) in R^2, a B-Manhattan network on T is a
network N(T) = (V,E) with the property that its edges are parallel to the
directions of B and for every pair of terminals t_i and t_j, the network N(T)
contains a shortest B-path between them, i.e., a path of length || t_i - t_j
||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum
possible length. The problem of finding minimum B-Manhattan networks has been
introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case
when the unit ball B is a square (and hence the distance || p - q || is the l_1
or the l_infty-distance between p and q) and it has been shown recently by
Chin, Guo, and Sun (SoCG'09) to be strongly NP-complete. Several approximation
algorithms (with factors 8, 4 ,3 , and 2) for minimum Manhattan problem are
known. In this paper, we propose a factor 2.5 approximation algorithm for
minimum B-Manhattan network problem. The algorithm employs a simplified version
of the strip-staircase decomposition proposed in our paper (APPROX'05) and
subsequently used in other factor 2 approximation algorithms for minimum
Manhattan problem.Comment: 16 pages, 5 figures; corrected typos, reference added, figure adde
Mean Staircase of the Riemann Zeros: a comment on the Lambert W function and an algebraic aspect
In this note we discuss explicitly the structure of two simple set of zeros
which are associated with the mean staircase emerging from the zeta function
and we specify a solution using the Lambert W function. The argument of it may
then be set equal to a special classical matrix (for every )
related to the Hamiltonian of the Mehta-Dyson model. In this way we specify a
function of an hermitean operator whose eigenvalues are the "trivial zeros" on
the critical line. The first set of trivial zeros is defined by the relations
\tmop{Im} (\zeta ({1/2} + i \cdot t)) = 0 \wedge \tmop{Re} (\zeta ({1/2} + i
\cdot t)) \neq 0 and viceversa for the second set. (To distinguish from the
usual trivial zeros , integer
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