402 research outputs found
Notes on highest weight modules of the elliptic algebra
We discuss a construction of highest weight modules for the recently defined
elliptic algebra , and make several conjectures
concerning them. The modules are generated by the action of the components of
the operator on the highest weight vectors. We introduce the vertex
operators and through their commutation relations with the
-operator. We present ordering rules for the - and -operators and
find an upper bound for the number of linearly independent vectors generated by
them, which agrees with the known characters of -modules.Comment: Nonstandard macro package eliminate
On characteristic points and approximate decision algorithms for the minimum Hausdorff distance
We investigate {\em approximate decision algorithms} for determining whether the minimum Hausdorff distance between two points sets (or between two sets of nonintersecting line segments) is at most .\def\eg{(\varepsilon/\gamma)} An approximate decision algorithm is a standard decision algorithm that answers {\sc yes} or {\sc no} except when is in an {\em indecision interval} where the algorithm is allowed to answer {\sc don't know}. We present algorithms with indecision interval where is the minimum Hausdorff distance and can be chosen by the user. In other words, we can make our algorithm as accurate as desired by choosing an appropriate . For two sets of points (or two sets of nonintersecting lines) with respective cardinalities and our approximate decision algorithms run in time O(\eg^2(m+n)\log(mn)) for Hausdorff distance under translation, and in time O(\eg^2mn\log(mn)) for Hausdorff distance under Euclidean motion
Conflict-Free Coloring Made Stronger
In FOCS 2002, Even et al. showed that any set of discs in the plane can
be Conflict-Free colored with a total of at most colors. That is,
it can be colored with colors such that for any (covered) point
there is some disc whose color is distinct from all other colors of discs
containing . They also showed that this bound is asymptotically tight. In
this paper we prove the following stronger results:
\begin{enumerate} \item [(i)] Any set of discs in the plane can be
colored with a total of at most colors such that (a) for any
point that is covered by at least discs, there are at least
distinct discs each of which is colored by a color distinct from all other
discs containing and (b) for any point covered by at most discs,
all discs covering are colored distinctively. We call such a coloring a
{\em -Strong Conflict-Free} coloring. We extend this result to pseudo-discs
and arbitrary regions with linear union-complexity.
\item [(ii)] More generally, for families of simple closed Jordan regions
with union-complexity bounded by , we prove that there exists
a -Strong Conflict-Free coloring with at most colors.
\item [(iii)] We prove that any set of axis-parallel rectangles can be
-Strong Conflict-Free colored with at most colors.
\item [(iv)] We provide a general framework for -Strong Conflict-Free
coloring arbitrary hypergraphs. This framework relates the notion of -Strong
Conflict-Free coloring and the recently studied notion of -colorful
coloring. \end{enumerate}
All of our proofs are constructive. That is, there exist polynomial time
algorithms for computing such colorings
Q-systems, Heaps, Paths and Cluster Positivity
We consider the cluster algebra associated to the -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -system are partition functions
for hard particles on particular target graphs with weights, which are
determined by the choice of initial data. This allows us to interpret the
simplest solutions of the Q-system as generating functions for Viennot's heaps
on these target graphs, and equivalently as generating functions of weighted
paths on suitable dual target graphs. The generating functions take the form of
finite continued fractions. In this setting, the cluster mutations correspond
to local rearrangements of the fractions which leave their final value
unchanged. Finally, the general solutions of the -system are interpreted as
partition functions for strongly non-intersecting families of lattice paths on
target lattices. This expresses all cluster variables as manifestly positive
Laurent polynomials of any initial data, thus proving the cluster positivity
conjecture for the -system. We also give an alternative formulation in
terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure
Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons
We consider the following motion-planning problem: we are given unit
discs in a simple polygon with vertices, each at their own start position,
and we want to move the discs to a given set of target positions. Contrary
to the standard (labeled) version of the problem, each disc is allowed to be
moved to any target position, as long as in the end every target position is
occupied. We show that this unlabeled version of the problem can be solved in
time, assuming that the start and target positions are at
least some minimal distance from each other. This is in sharp contrast to the
standard (labeled) and more general multi-robot motion-planning problem for
discs moving in a simple polygon, which is known to be strongly NP-hard
Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation
We construct solutions of an infinite Toda system and an analogue of the
Painlev'e II equation over noncommutative differential division rings in terms
of quasideterminants of Hankel matrices.Comment: 16 pp; final revised version, will appear in J.Phys. A, minor changes
(typos corrected following the Referee's List, aknowledgements and a new
reference added
Spectral Equivalence of Bosons and Fermions in One-Dimensional Harmonic Potentials
Recently, Schmidt and Schnack (cond-mat/9803151, cond-mat/9810036), following
earlier references, reiterate that the specific heat of N non-interacting
bosons in a one-dimensional harmonic well equals that of N fermions in the same
potential. We show that this peculiar relationship between specific heats
results from a more dramatic equivalence between bose and fermi systems.
Namely, we prove that the excitation spectrums of such bose and fermi systems
are spectrally equivalent. Two complementary proofs are provided, one based on
an analysis of the dynamical symmetry group of the N-body system, the other
using combinatoric analysis.Comment: Six Pages, No Figures, Submitted to Phys. Rev.
Spinons in Magnetic Chains of Arbitrary Spins at Finite Temperatures
The thermodynamics of solvable isotropic chains with arbitrary spins is
addressed by the recently developed quantum transfer matrix (QTM) approach. The
set of nonlinear equations which exactly characterize the free energy is
derived by respecting the physical excitations at T=0, spinons and RSOS kinks.
We argue the implication of the present formulation to spinon character formula
of level k=2S SU(2) WZWN model .Comment: 25 pages, 8 Postscript figures, Latex 2e,uses graphicx, added figures
and detailed discussion
Spinons and parafermions in fermion cosets
We introduce a set of gauge invariant fermion fields in fermionic coset
models and show that they play a very central role in the description of
several Conformal Field Theories (CFT's). In particular we discuss the explicit
realization of primaries and their OPE in unitary minimal models, parafermion
fields in CFT's and that of spinon fields in
Wess-Zumino-Witten models (WZW) theories. The higher level case () will be
briefly discussed. Possible applications to QHE systems and spin-ladder systems
are addressed.Comment: 6 pages, Latex file. Invited talk at International Seminar dedicated
to the memory of D.V.Volkov, Kharkov, January 5-7, 199
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