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Approximating parabolas as natural bounds of Heisenberg spectra: Reply on the comment of O. Waldmann
O. Waldmann has shown that some spin systems, which fulfill the condition of
a weakly homogeneous coupling matrix, have a spectrum whose minimal or maximal
energies are rather poorly approximated by a quadratic dependence on the total
spin quantum number. We comment on this observation and provide the new
argument that, under certain conditions, the approximating parabolas appear as
natural bounds of the spectrum generated by spin coherent states.Comment: 2 pages, accepted for Europhysics Letter
Context unification is in PSPACE
Contexts are terms with one `hole', i.e. a place in which we can substitute
an argument. In context unification we are given an equation over terms with
variables representing contexts and ask about the satisfiability of this
equation. Context unification is a natural subvariant of second-order
unification, which is undecidable, and a generalization of word equations,
which are decidable, at the same time. It is the unique problem between those
two whose decidability is uncertain (for already almost two decades). In this
paper we show that the context unification is in PSPACE. The result holds under
a (usual) assumption that the first-order signature is finite.
This result is obtained by an extension of the recompression technique,
recently developed by the author and used in particular to obtain a new PSPACE
algorithm for satisfiability of word equations, to context unification. The
recompression is based on performing simple compression rules (replacing pairs
of neighbouring function symbols), which are (conceptually) applied on the
solution of the context equation and modifying the equation in a way so that
such compression steps can be in fact performed directly on the equation,
without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements
over the previous tex
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