56,429 research outputs found
Coherent state triplets and their inner products
It is shown that if H is a Hilbert space for a representation of a group G,
then there are triplets of spaces F_H, H, F^H, in which F^H is a space of
coherent state or vector coherent state wave functions and F_H is its dual
relative to a conveniently defined measure. It is shown also that there is a
sequence of maps F_H -> H -> F^H which facilitates the construction of the
corresponding inner products. After completion if necessary, the F_H, H, and
F^H, become isomorphic Hilbert spaces. It is shown that the inner product for H
is often easier to evaluate in F_H than F^H. Thus, we obtain integral
expressions for the inner products of coherent state and vector coherent state
representations. These expressions are equivalent to the algebraic expressions
of K-matrix theory, but they are frequently more efficient to apply. The
construction is illustrated by many examples.Comment: 33 pages, RevTex (Latex2.09) This paper is withdrawn because it
contained errors that are being correcte
Algebraic-matrix calculation of vibrational levels of triatomic molecules
We introduce an accurate and efficient algebraic technique for the
computation of the vibrational spectra of triatomic molecules, of both linear
and bent equilibrium geometry. The full three-dimensional potential energy
surface (PES), which can be based on entirely {\it ab initio} data, is
parameterized as a product Morse-cosine expansion, expressed in bond-angle
internal coordinates, and includes explicit interactions among the local modes.
We describe the stretching degrees of freedom in the framework of a Morse-type
expansion on a suitable algebraic basis, which provides exact analytical
expressions for the elements of a sparse Hamiltonian matrix. Likewise, we use a
cosine power expansion on a spherical harmonics basis for the bending degree of
freedom. The resulting matrix representation in the product space is very
sparse and vibrational levels and eigenfunctions can be obtained by efficient
diagonalization techniques. We apply this method to carbonyl sulfide OCS,
hydrogen cyanide HCN, water HO, and nitrogen dioxide NO. When we base
our calculations on high-quality PESs tuned to the experimental data, the
computed spectra are in very good agreement with the observed band origins.Comment: 11 pages, 2 figures, containg additional supporting information in
epaps.ps (results in tables, which are useful but not too important for the
paper
Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the
\emph{Algebraic Eraser} scheme for key agreement over an insecure channel,
using a novel hybrid of infinite and finite noncommutative groups. They also
introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete
realization of this scheme.
We present general, efficient heuristic algorithms, which extract the shared
key out of the public information provided by CBKAP. These algorithms are,
according to heuristic reasoning and according to massive experiments,
successful for all sizes of the security parameters, assuming that the keys are
chosen with standard distributions.
Our methods come from probabilistic group theory (permutation group actions
and expander graphs). In particular, we provide a simple algorithm for finding
short expressions of permutations in , as products of given random
permutations. Heuristically, our algorithm gives expressions of length
, in time and space . Moreover, this is provable from
\emph{the Minimal Cycle Conjecture}, a simply stated hypothesis concerning the
uniform distribution on . Experiments show that the constants in these
estimations are small. This is the first practical algorithm for this problem
for .
Remark: \emph{Algebraic Eraser} is a trademark of SecureRF. The variant of
CBKAP actually implemented by SecureRF uses proprietary distributions, and thus
our results do not imply its vulnerability. See also arXiv:abs/12020598Comment: Final version, accepted to Advances in Applied Mathematics. Title
slightly change
Algebraic optimization of recursive queries
Over the past few years, much attention has been paid to deductive databases. They offer a logic-based interface, and allow formulation of complex recursive queries. However, they do not offer appropriate update facilities, and do not support existing applications. To overcome these problems an SQL-like interface is required besides a logic-based interface.\ud
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In the PRISMA project we have developed a tightly-coupled distributed database, on a multiprocessor machine, with two user interfaces: SQL and PRISMAlog. Query optimization is localized in one component: the relational query optimizer. Therefore, we have defined an eXtended Relational Algebra that allows recursive query formulation and can also be used for expressing executable schedules, and we have developed algebraic optimization strategies for recursive queries. In this paper we describe an optimization strategy that rewrites regular (in the context of formal grammars) mutually recursive queries into standard Relational Algebra and transitive closure operations. We also describe how to push selections into the resulting transitive closure operations.\ud
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The reason we focus on algebraic optimization is that, in our opinion, the new generation of advanced database systems will be built starting from existing state-of-the-art relational technology, instead of building a completely new class of systems
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