201 research outputs found
Vertices of Gelfand-Tsetlin Polytopes
This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns
arising in the representation theory \mathfrak{gl}_n \C and algebraic
combinatorics. We present a combinatorial characterization of the vertices and
a method to calculate the dimension of the lowest-dimensional face containing a
given Gelfand-Tsetlin pattern.
As an application, we disprove a conjecture of Berenstein and Kirillov about
the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can
construct for each a counterexample, with arbitrarily increasing
denominators as grows, of a non-integral vertex. This is the first infinite
family of non-integral polyhedra for which the Ehrhart counting function is
still a polynomial. We also derive a bound on the denominators for the
non-integral vertices when is fixed.Comment: 14 pages, 3 figures, fixed attribution
On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect
We investigate the problem of computing tensor product multiplicities for
complex semisimple Lie algebras. Even though computing these numbers is #P-hard
in general, we show that if the rank of the Lie algebra is assumed fixed, then
there is a polynomial time algorithm, based on counting the lattice points in
polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based
on the ellipsoid algorithm, to decide when the coefficients are nonzero in
polynomial time for arbitrary rank. Our experiments show that the lattice point
algorithm is superior in practice to the standard techniques for computing
multiplicities when the weights have large entries but small rank. Using an
implementation of this algorithm, we provide experimental evidence for
conjectured generalizations of the saturation property of
Littlewood--Richardson coefficients. One of these conjectures seems to be valid
for types B_n, C_n, and D_n.Comment: 21 pages, 6 table
Different types of integrability and their relation to decoherence in central spin models
We investigate the relation between integrability and decoherence in central
spin models with more than one central spin. We show that there is a transition
between integrability ensured by the Bethe ansatz and integrability ensured by
complete sets of commuting operators. This has a significant impact on the
decoherence properties of the system, suggesting that it is not necessarily
integrability or nonintegrability which is related to decoherence, but rather
its type or a change from integrability to nonintegrability.Comment: 4 pages, 3 figure
Shortening the Frommelt Attitude Toward the Care Of the Dying Scale (FATCOD-B): a Brief 9-Item Version for Medical Education and Practice
End-of-life care training has gaps in helping students to develop attitudes toward caring for the dying. Valid and reliable assessment tools are essential in building effective educational programmes. The Frommelt Attitude Toward the Care Of the Dying scale (FATCOD-B) is widely used to measure the level of comfort/discomfort in caring for the dying and to test the effectiveness of end-of-life care training. However, its psychometric properties have been questioned and different proposals for refinement and shortening have been put forward. The aim of this study is to get to a definitive reduction of the FATCOD-B through a valid and parsimonious synthesis of the previous attempts at scale revision. Data were gathered from a sample of 220 medical students. The item response theory approach was used in this study. Of the 14 items selected from two previous proposals for scale revision, 3 had a weak correlation with the whole scale and were deleted. The resulting 11-item version had good fit indices and withstood a more general and parsimonious specification (rating scale model). This solution was further shortened to 9 items by deleting 2 of 3 items at the same level of difficulty. The final 9-item version was invariant for gender, level of religiosity and amount of experience with dying persons, free from redundant items and able to scale and discriminate the respondents
Equivalence of domains for hyperbolic Hubbard-Stratonovich transformations
We settle a long standing issue concerning the traditional derivation of
non-compact non-linear sigma models in the theory of disordered electron
systems: the hyperbolic Hubbard-Stratonovich (HS) transformation of
Pruisken-Schaefer type. Only recently the validity of such transformations was
proved in the case of U(p,q) (non-compact unitary) and O(p,q) (non-compact
orthogonal) symmetry. In this article we give a proof for general non-compact
symmetry groups. Moreover we show that the Pruisken-Schaefer type
transformations are related to other variants of the HS transformation by
deformation of the domain of integration. In particular we clarify the origin
of surprising sign factors which were recently discovered in the case of
orthogonal symmetry.Comment: 30 pages, 3 figure
A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope
We present a multivariate generating function for all n x n nonnegative
integral matrices with all row and column sums equal to a positive integer t,
the so called semi-magic squares. As a consequence we obtain formulas for all
coefficients of the Ehrhart polynomial of the polytope B_n of n x n
doubly-stochastic matrices, also known as the Birkhoff polytope. In particular
we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric
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