14,854 research outputs found
Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions
The -Young lattice is a partial order on partitions with no part
larger than . This weak subposet of the Young lattice originated from the
study of the -Schur functions(atoms) , symmetric functions
that form a natural basis of the space spanned by homogeneous functions indexed
by -bounded partitions. The chains in the -Young lattice are induced by a
Pieri-type rule experimentally satisfied by the -Schur functions. Here,
using a natural bijection between -bounded partitions and -cores, we
establish an algorithm for identifying chains in the -Young lattice with
certain tableaux on cores. This algorithm reveals that the -Young
lattice is isomorphic to the weak order on the quotient of the affine symmetric
group by a maximal parabolic subgroup. From this, the
conjectured -Pieri rule implies that the -Kostka matrix connecting the
homogeneous basis \{h_\la\}_{\la\in\CY^k} to \{s_\la^{(k)}\}_{\la\in\CY^k}
may now be obtained by counting appropriate classes of tableaux on -cores.
This suggests that the conjecturally positive -Schur expansion coefficients
for Macdonald polynomials (reducing to -Kostka polynomials for large )
could be described by a -statistic on these tableaux, or equivalently on
reduced words for affine permutations.Comment: 30 pages, 1 figur
A Hybrid Observer for a Distributed Linear System with a Changing Neighbor Graph
A hybrid observer is described for estimating the state of an channel,
-dimensional, continuous-time, distributed linear system of the form
. The system's state is
simultaneously estimated by agents assuming each agent senses and
receives appropriately defined data from each of its current neighbors.
Neighbor relations are characterized by a time-varying directed graph
whose vertices correspond to agents and whose arcs depict
neighbor relations. Agent updates its estimate of at "event
times" using a local observer and a local parameter
estimator. The local observer is a continuous time linear system whose input is
and whose output is an asymptotically correct estimate of
where a matrix with kernel equaling the unobservable space of .
The local parameter estimator is a recursive algorithm designed to estimate,
prior to each event time , a constant parameter which satisfies the
linear equations , where is a small
positive constant and is the state estimation error of local observer
. Agent accomplishes this by iterating its parameter estimator state
, times within the interval , and by making use of
the state of each of its neighbors' parameter estimators at each iteration. The
updated value of at event time is then . Subject to the assumptions that (i) the neighbor graph
is strongly connected for all time, (ii) the system whose state
is to be estimated is jointly observable, (iii) is sufficiently large, it
is shown that each estimate converges to exponentially fast as
at a rate which can be controlled.Comment: 7 pages, the 56th IEEE Conference on Decision and Contro
Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness
Studying the problem of wave propagation in media with resistive boundaries
can be made by searching for "resonance modes" or free oscillations regimes. In
the present article, a simple case is investigated, which allows one to
enlighten the respective interest of different, classical methods, some of them
being rather delicate. This case is the 1D propagation in a homogeneous medium
having two purely resistive terminations, the calculation of the Green function
being done without any approximation using three methods. The first one is the
straightforward use of the closed-form solution in the frequency domain and the
residue calculus. Then the method of separation of variables (space and time)
leads to a solution depending on the initial conditions. The question of the
orthogonality and completeness of the complex-valued resonance modes is
investigated, leading to the expression of a particular scalar product. The
last method is the expansion in biorthogonal modes in the frequency domain, the
modes having eigenfrequencies depending on the frequency. Results of the three
methods generalize or/and correct some results already existing in the
literature, and exhibit the particular difficulty of the treatment of the
constant mode
An interacting quark-diquark model of baryons
A simple quark-diquark model of baryons with direct and exchange interactions
is constructed. Spectrum and form factors are calculated and compared with
experimental data. Advantages and disadvantages of the model are discussed.Comment: 13 pages, 3 eps-figures, accepted by Phys.Rev. C Rapid Communication
Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results
We develop powerful numerical and analytical techniques for the solution of
the Helmholtz equation on general domains. We prove two theorems: the first
theorem provides an exact formula for the ground state of an arbirtrary
membrane, while the second theorem generalizes this result to any excited state
of the membrane. We also develop a systematic perturbative scheme which can be
used to study the small deformations of a membrane of circular or square
shapes. We discuss several applications, obtaining numerical and analytical
results.Comment: 29 pages, 12 figures, 7 tabl
An affine approach to Peterson comparison
The Peterson comparison formula proved by Woodward relates the three-pointedGromov-Witten invariants for the quantum cohomology of partial flag varietiesto those for the complete flag. Another such comparison can be obtained bycomposing a combinatorial version of the Peterson isomorphism with a result ofLapointe and Morse relating quantum Littlewood-Richardson coefficients for theGrassmannian to k-Schur analogs in the homology of the affine Grassmannianobtained by adding rim hooks. We show that these comparisons on quantumcohomology are equivalent, up to Postnikov's strange duality isomorphism.<br
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