265,671 research outputs found
A strong operator topology adiabatic theorem
We prove an adiabatic theorem for the evolution of spectral data under a weak
additive perturbation in the context of a system without an intrinsic time
scale. For continuous functions of the unperturbed Hamiltonian the convergence
is in norm while for a larger class functions, including the spectral
projections associated to embedded eigenvalues, the convergence is in the
strong operator topology.Comment: 15 pages, no figure
Adaptive Gibbs samplers
We consider various versions of adaptive Gibbs and Metropolis-
within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the
fly during a run, by learning
as they go in an attempt to optimise the algorithm. We present a cautionary
example of how even a simple-seeming adaptive Gibbs sampler may fail to
converge. We then present various positive results guaranteeing convergence
of adaptive Gibbs samplers under certain conditions
Generating functions for Wilf equivalence under generalized factor order
Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on
words comprised of letters from a partially ordered set by
setting if there is a subword of of the same length as
such that the -th character of is greater than or equal to the -th
character of for all . This subword is called an embedding of
into . For the case where is the positive integers with the usual
ordering, they defined the weight of a word to be
, and the corresponding weight
generating function . They then
defined two words and to be Wilf equivalent, denoted , if
and only if . They also defined the related generating
function where
is the set of all words such that the only embedding of
into is a suffix of , and showed that if and only if
. We continue this study by giving an explicit formula for
if factors into a weakly increasing word followed by a weakly
decreasing word. We use this formula as an aid to classify Wilf equivalence for
all words of length 3. We also show that coefficients of related generating
functions are well-known sequences in several special cases. Finally, we
discuss a conjecture that if then and must be
rearrangements, and the stronger conjecture that there also must be a
weight-preserving bijection such
that is a rearrangement of for all .Comment: 23 page
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