635,273 research outputs found
Metropolis Integration Schemes for Self-Adjoint Diffusions
We present explicit methods for simulating diffusions whose generator is
self-adjoint with respect to a known (but possibly not normalizable) density.
These methods exploit this property and combine an optimized Runge-Kutta
algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting
numerical integration scheme is shown to be weakly accurate at finite noise and
to gain higher order accuracy in the small noise limit. It also permits to
avoid computing explicitly certain terms in the equation, such as the
divergence of the mobility tensor, which can be tedious to calculate. Finally,
the scheme is shown to be ergodic with respect to the exact equilibrium
probability distribution of the diffusion when it exists. These results are
illustrated on several examples including a Brownian dynamics simulation of DNA
in a solvent. In this example, the proposed scheme is able to accurately
compute dynamics at time step sizes that are an order of magnitude (or more)
larger than those permitted with commonly used explicit predictor-corrector
schemes.Comment: 54 pages, 8 figures, To appear in MM
Static Data Structure Lower Bounds Imply Rigidity
We show that static data structure lower bounds in the group (linear) model
imply semi-explicit lower bounds on matrix rigidity. In particular, we prove
that an explicit lower bound of on the cell-probe
complexity of linear data structures in the group model, even against
arbitrarily small linear space , would already imply a
semi-explicit () construction of rigid matrices with
significantly better parameters than the current state of art (Alon, Panigrahy
and Yekhanin, 2009). Our results further assert that polynomial () data structure lower bounds against near-optimal space, would
imply super-linear circuit lower bounds for log-depth linear circuits (a
four-decade open question). In the succinct space regime , we show
that any improvement on current cell-probe lower bounds in the linear model
would also imply new rigidity bounds. Our results rely on a new connection
between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak,
2006), and on a new reduction from worst-case to average-case rigidity, which
is of independent interest
On the Hierarchy of Block Deterministic Languages
A regular language is -lookahead deterministic (resp. -block
deterministic) if it is specified by a -lookahead deterministic (resp.
-block deterministic) regular expression. These two subclasses of regular
languages have been respectively introduced by Han and Wood (-lookahead
determinism) and by Giammarresi et al. (-block determinism) as a possible
extension of one-unambiguous languages defined and characterized by
Br\"uggemann-Klein and Wood. In this paper, we study the hierarchy and the
inclusion links of these families. We first show that each -block
deterministic language is the alphabetic image of some one-unambiguous
language. Moreover, we show that the conversion from a minimal DFA of a
-block deterministic regular language to a -block deterministic automaton
not only requires state elimination, and that the proof given by Han and Wood
of a proper hierarchy in -block deterministic languages based on this result
is erroneous. Despite these results, we show by giving a parameterized family
that there is a proper hierarchy in -block deterministic regular languages.
We also prove that there is a proper hierarchy in -lookahead deterministic
regular languages by studying particular properties of unary regular
expressions. Finally, using our valid results, we confirm that the family of
-block deterministic regular languages is strictly included into the one of
-lookahead deterministic regular languages by showing that any -block
deterministic unary language is one-unambiguous
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