1,053 research outputs found
Quantum Knizhnik-Zamolodchikov Equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices
We present multiresidue formulae for partial sums in the basis of link
patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum
Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q.
These allow for rewriting and generalizing a recent conjecture [Di Francesco
'06] connecting the above to generating polynomials for weighted Totally
Symmetric Self-Complementary Plane Partitions. We reduce the corresponding
conjectures to a single integral identity, yet to be proved
Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
We propose new conjectures relating sum rules for the polynomial solution of
the qKZ equation with open (reflecting) boundaries as a function of the quantum
parameter and the -enumeration of Plane Partitions with specific
symmetries, with . We also find a conjectural relation \`a la
Razumov-Stroganov between the limit of the qKZ solution and refined
numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision
TAPER: query-aware, partition-enhancement for large, heterogenous, graphs
Graph partitioning has long been seen as a viable approach to address Graph
DBMS scalability. A partitioning, however, may introduce extra query processing
latency unless it is sensitive to a specific query workload, and optimised to
minimise inter-partition traversals for that workload. Additionally, it should
also be possible to incrementally adjust the partitioning in reaction to
changes in the graph topology, the query workload, or both. Because of their
complexity, current partitioning algorithms fall short of one or both of these
requirements, as they are designed for offline use and as one-off operations.
The TAPER system aims to address both requirements, whilst leveraging existing
partitioning algorithms. TAPER takes any given initial partitioning as a
starting point, and iteratively adjusts it by swapping chosen vertices across
partitions, heuristically reducing the probability of inter-partition
traversals for a given pattern matching queries workload. Iterations are
inexpensive thanks to time and space optimisations in the underlying support
data structures. We evaluate TAPER on two different large test graphs and over
realistic query workloads. Our results indicate that, given a hash-based
partitioning, TAPER reduces the number of inter-partition traversals by around
80%; given an unweighted METIS partitioning, by around 30%. These reductions
are achieved within 8 iterations and with the additional advantage of being
workload-aware and usable online.Comment: 12 pages, 11 figures, unpublishe
Geometry and symmetries of multi-particle systems
The quantum dynamical evolution of atomic and molecular aggregates, from
their compact to their fragmented states, is parametrized by a single
collective radial parameter. Treating all the remaining particle coordinates in
d dimensions democratically, as a set of angles orthogonal to this collective
radius or by equivalent variables, bypasses all independent-particle
approximations. The invariance of the total kinetic energy under arbitrary
d-dimensional transformations which preserve the radial parameter gives rise to
novel quantum numbers and ladder operators interconnecting its eigenstates at
each value of the radial parameter.
We develop the systematics and technology of this approach, introducing the
relevant mathematics tutorially, by analogy to the familiar theory of angular
momentum in three dimensions. The angular basis functions so obtained are
treated in a manifestly coordinate-free manner, thus serving as a flexible
generalized basis for carrying out detailed studies of wavefunction evolution
in multi-particle systems.Comment: 37 pages, 2 eps figure
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