51,728 research outputs found
q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers
We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi–Stirling numbers.
This study is motivated by their key role in the (reciprocal) expansion of any power of a second order
q-differential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators,
which we explicitly construct in this work. The results here obtained can be viewed as the q-version of
those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a
q-version of the Jacobi–Stirling numbers given by Gelineau and the second author
Ab Initio Simulation of the Nodal Surfaces of Heisenberg Antiferromagnets
The spin-half Heisenberg antiferromagnet (HAF) on the square and triangular
lattices is studied using the coupled cluster method (CCM) technique of quantum
many-body theory. The phase relations between different expansion coefficients
of the ground-state wave function in an Ising basis for the square lattice HAF
is exactly known via the Marshall-Peierls sign rule, although no equivalent
sign rule has yet been obtained for the triangular lattice HAF. Here the CCM is
used to give accurate estimates for the Ising-expansion coefficients for these
systems, and CCM results are noted to be fully consistent with the
Marshall-Peierls sign rule for the square lattice case. For the triangular
lattice HAF, a heuristic rule is presented which fits our CCM results for the
Ising-expansion coefficients of states which correspond to two-body excitations
with respect to the reference state. It is also seen that Ising-expansion
coefficients which describe localised, -body excitations with respect to the
reference state are found to be highly converged, and from this result we infer
that the nodal surface of the triangular lattice HAF is being accurately
modeled. Using these results, we are able to make suggestions regarding
possible extensions of existing quantum Monte Carlo simulations for the
triangular lattice HAF.Comment: 24 pages, Latex, 3 postscript figure
Optimal social security in a dynastic model with investment externalities and endogenous fertility
This paper studies optimal pay-as-you-go social security with positive bequests and endogenous fertility. With an investment externality, a competitive solution without social security su?ers from under-investment in capital and over-reproduction of population. We show that social security can improve welfare by reducing fertility and increasing capital intensity. We also illustrate numerically that a small degree of this externality is enough to justify the observed high ratios of social security spending to GDP.
On Zudilin's q-question about Schmidt's problem
We propose an elemantary approach to Zudilin's q-question about Schmidt's
problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a
previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a
q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740]
derived from q-Pfaff-Saalschutz identity.Comment: 5 page
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