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-modules, Bernstein-Sato polynomials and -invariants of direct summands
We study the structure of -modules over a ring which is a direct
summand of a polynomial or a power series ring with coefficients over a
field. We relate properties of -modules over to -modules over . We
show that the localization and the local cohomology module
have finite length as -modules over . Furthermore, we show the existence
of the Bernstein-Sato polynomial for elements in . In positive
characteristic, we use this relation between -modules over and to
show that the set of -jumping numbers of an ideal is
contained in the set of -jumping numbers of its extension in . As a
consequence, the -jumping numbers of in form a discrete set of
rational numbers. We also relate the Bernstein-Sato polynomial in with the
-thresholds and the -jumping numbers in .Comment: 24 pages. Comments welcome
An extension of the coupled-cluster method: A variational formalism
A general quantum many-body theory in configuration space is developed by
extending the traditional coupled cluter method (CCM) to a variational
formalism. Two independent sets of distribution functions are introduced to
evaluate the Hamiltonian expectation. An algebraic technique for calculating
these distribution functions via two self-consistent sets of equations is
given. By comparing with the traditional CCM and with Arponen's extension, it
is shown that the former is equivalent to a linear approximation to one set of
distribution functions and the later is equivalent to a random-phase
approximation to it. In additional to these two approximations, other
higher-order approximation schemes within the new formalism are also discussed.
As a demonstration, we apply this technique to a quantum antiferromagnetic spin
model.Comment: 15 pages. Submitted to Phys. Rev.
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