996 research outputs found
Statistical Complexity of Simple 1D Spin Systems
We present exact results for two complementary measures of spatial structure
generated by 1D spin systems with finite-range interactions. The first, excess
entropy, measures the apparent spatial memory stored in configurations. The
second, statistical complexity, measures the amount of memory needed to
optimally predict the chain of spin values. These statistics capture distinct
properties and are different from existing thermodynamic quantities.Comment: 4 pages with 2 eps Figures. Uses RevTeX macros. Also available at
http://www.santafe.edu/projects/CompMech/papers/CompMechCommun.htm
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
Exact Synchronization for Finite-State Sources
We analyze how an observer synchronizes to the internal state of a
finite-state information source, using the epsilon-machine causal
representation. Here, we treat the case of exact synchronization, when it is
possible for the observer to synchronize completely after a finite number of
observations. The more difficult case of strictly asymptotic synchronization is
treated in a sequel. In both cases, we find that an observer, on average, will
synchronize to the source state exponentially fast and that, as a result, the
average accuracy in an observer's predictions of the source output approaches
its optimal level exponentially fast as well. Additionally, we show here how to
analytically calculate the synchronization rate for exact epsilon-machines and
provide an efficient polynomial-time algorithm to test epsilon-machines for
exactness.Comment: 9 pages, 6 figures; now includes analytical calculation of the
synchronization rate; updates and corrections adde
MODELING THE COSTS OF FOOD SAFETY REGULATION
Food safety, regulatory costs, cost/benefit analysis, Food Consumption/Nutrition/Food Safety,
Diffraction Patterns of Layered Close-packed Structures from Hidden Markov Models
We recently derived analytical expressions for the pairwise (auto)correlation
functions (CFs) between modular layers (MLs) in close-packed structures (CPSs)
for the wide class of stacking processes describable as hidden Markov models
(HMMs) [Riechers \etal, (2014), Acta Crystallogr.~A, XX 000-000]. We now use
these results to calculate diffraction patterns (DPs) directly from HMMs,
discovering that the relationship between the HMMs and DPs is both simple and
fundamental in nature. We show that in the limit of large crystals, the DP is a
function of parameters that specify the HMM. We give three elementary but
important examples that demonstrate this result, deriving expressions for the
DP of CPSs stacked (i) independently, (ii) as infinite-Markov-order randomly
faulted 2H and 3C stacking structures over the entire range of growth and
deformation faulting probabilities, and (iii) as a HMM that models
Shockley-Frank stacking faults in 6H-SiC. While applied here to planar faulting
in CPSs, extending the methods and results to planar disorder in other layered
materials is straightforward. In this way, we effectively solve the broad
problem of calculating a DP---either analytically or numerically---for any
stacking structure---ordered or disordered---where the stacking process can be
expressed as a HMM.Comment: 18 pages, 6 figures, 3 tables;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dplcps.ht
Predicting subjective workload ratings: A comparison and synthesis of theoretical models.
Output data from a computer simulation of two air traffic control (ATC) scenarios were fit to workload ratings that ATC subject matter experts provided while observing each scenario in real time. Simulation output enabled regressions to test the assumptions of a variety of workload prediction models. The models included operational models that use observable situational and behavior variables (such as number of aircraft and communications by type) and theoretical models that use queuing and cognitive architecture variables (such as weightings of activities performed, amount of busy time, and sensory and cognitive resource usage) to predict workload. Regression results suggest models that include number of activities performed weighted by priority are best able to account for the highest amount of variance in subjective workload ratings
Reductions of Hidden Information Sources
In all but special circumstances, measurements of time-dependent processes
reflect internal structures and correlations only indirectly. Building
predictive models of such hidden information sources requires discovering, in
some way, the internal states and mechanisms. Unfortunately, there are often
many possible models that are observationally equivalent. Here we show that the
situation is not as arbitrary as one would think. We show that generators of
hidden stochastic processes can be reduced to a minimal form and compare this
reduced representation to that provided by computational mechanics--the
epsilon-machine. On the way to developing deeper, measure-theoretic foundations
for the latter, we introduce a new two-step reduction process. The first step
(internal-event reduction) produces the smallest observationally equivalent
sigma-algebra and the second (internal-state reduction) removes sigma-algebra
components that are redundant for optimal prediction. For several classes of
stochastic dynamical systems these reductions produce representations that are
equivalent to epsilon-machines.Comment: 12 pages, 4 figures; 30 citations; Updates at
http://www.santafe.edu/~cm
A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression
The causal structure of a stochastic process can be more efficiently
transmitted via a quantum channel than a classical one, an advantage that
increases with codeword length. While previously difficult to compute, we
express the quantum advantage in closed form using spectral decomposition,
leading to direct computation of the quantum communication cost at all encoding
lengths, including infinite. This makes clear how finite-codeword compression
is controlled by the classical process' cryptic order and allows us to analyze
structure within the length-asymptotic regime of infinite-cryptic order (and
infinite Markov order) processes.Comment: 21 pages, 13 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht
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