7,516 research outputs found
Exact Probability Distribution versus Entropy
The problem addressed concerns the determination of the average number of
successive attempts of guessing a word of a certain length consisting of
letters with given probabilities of occurrence. Both first- and second-order
approximations to a natural language are considered. The guessing strategy used
is guessing words in decreasing order of probability. When word and alphabet
sizes are large, approximations are necessary in order to estimate the number
of guesses. Several kinds of approximations are discussed demonstrating
moderate requirements concerning both memory and CPU time. When considering
realistic sizes of alphabets and words (100) the number of guesses can be
estimated within minutes with reasonable accuracy (a few percent). For many
probability distributions the density of the logarithm of probability products
is close to a normal distribution. For those cases it is possible to derive an
analytical expression for the average number of guesses. The proportion of
guesses needed on average compared to the total number decreases almost
exponentially with the word length. The leading term in an asymptotic expansion
can be used to estimate the number of guesses for large word lengths.
Comparisons with analytical lower bounds and entropy expressions are also
provided
Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns
Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is
described by means of an infinite hierarchy of its unstable spatiotemporally
periodic solutions. An intrinsic parametrization of the corresponding invariant
set serves as accurate guide to the high-dimensional dynamics, and the periodic
orbit theory yields several global averages characterizing the chaotic
dynamics.Comment: Latex, ioplppt.sty and iopl10.sty, 18 pages, 11 PS-figures,
compressed and encoded with uufiles, 170 k
A Doubly Nudged Elastic Band Method for Finding Transition States
A modification of the nudged elastic band (NEB) method is presented that
enables stable optimisations to be run using both the limited-memory
quasi-Newton (L-BFGS) and slow-response quenched velocity Verlet (SQVV)
minimisers. The performance of this new `doubly nudged' DNEB method is analysed
in conjunction with both minimisers and compared with previous NEB
formulations. We find that the fastest DNEB approach (DNEB/L-BFGS) can be
quicker by up to two orders of magnitude. Applications to permutational
rearrangements of the seven-atom Lennard-Jones cluster (LJ7) and highly
cooperative rearrangements of LJ38 and LJ75 are presented. We also outline an
updated algorithm for constructing complicated multi-step pathways using
successive DNEB runs.Comment: 13 pages, 8 figures, 2 table
On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation
In this paper we address the stable numerical solution of nonlinear ill-posed
systems by a trust-region method. We show that an appropriate choice of the
trust-region radius gives rise to a procedure that has the potential to
approach a solution of the unperturbed system. This regularizing property is
shown theoretically and validated numerically.Comment: arXiv admin note: text overlap with arXiv:1410.278
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