564,896 research outputs found
Criticality, the area law, and the computational power of PEPS
The projected entangled pair state (PEPS) representation of quantum states on
two-dimensional lattices induces an entanglement based hierarchy in state
space. We show that the lowest levels of this hierarchy exhibit an enormously
rich structure including states with critical and topological properties as
well as resonating valence bond states. We prove, in particular, that coherent
versions of thermal states of any local 2D classical spin model correspond to
such PEPS, which are in turn ground states of local 2D quantum Hamiltonians.
This correspondence maps thermal onto quantum fluctuations, and it allows us to
analytically construct critical quantum models exhibiting a strict area law
scaling of the entanglement entropy in the face of power law decaying
correlations. Moreover, it enables us to show that there exist PEPS within the
same class as the cluster state, which can serve as computational resources for
the solution of NP-hard problems
Adaptive Event Horizon Tracking and Critical Phenomena in Binary Black Hole Coalescence
This work establishes critical phenomena in the topological transition of
black hole coalescence. We describe and validate a computational front tracking
event horizon solver, developed for generic studies of the black hole
coalescence problem. We then apply this to the Kastor - Traschen axisymmetric
analytic solution of the extremal Maxwell - Einstein black hole merger with
cosmological constant. The surprising result of this computational analysis is
a power law scaling of the minimal throat proportional to time. The minimal
throat connecting the two holes obeys this power law during a short time
immediately at the beginning of merger. We also confirm the behavior
analytically. Thus, at least in one axisymmetric situation a critical
phenomenon exists. We give arguments for a broader universality class than the
restricted requirements of the Kastor - Traschen solution.Comment: 13 pages, 20 figures Corrected labels on figures 17 through 20.
Corrected typos in references. Added some comment
A multi-domain hybrid method for head-on collision of black holes in particle limit
A hybrid method is developed based on the spectral and finite-difference
methods for solving the inhomogeneous Zerilli equation in time-domain. The
developed hybrid method decomposes the domain into the spectral and
finite-difference domains. The singular source term is located in the spectral
domain while the solution in the region without the singular term is
approximated by the higher-order finite-difference method.
The spectral domain is also split into multi-domains and the
finite-difference domain is placed as the boundary domain. Due to the global
nature of the spectral method, a multi-domain method composed of the spectral
domains only does not yield the proper power-law decay unless the range of the
computational domain is large. The finite-difference domain helps reduce
boundary effects due to the truncation of the computational domain. The
multi-domain approach with the finite-difference boundary domain method reduces
the computational costs significantly and also yields the proper power-law
decay.
Stable and accurate interface conditions between the finite-difference and
spectral domains and the spectral and spectral domains are derived. For the
singular source term, we use both the Gaussian model with various values of
full width at half maximum and a localized discrete -function. The
discrete -function was generalized to adopt the Gauss-Lobatto
collocation points of the spectral domain.
The gravitational waveforms are measured. Numerical results show that the
developed hybrid method accurately yields the quasi-normal modes and the
power-law decay profile. The numerical results also show that the power-law
decay profile is less sensitive to the shape of the regularized
-function for the Gaussian model than expected. The Gaussian model also
yields better results than the localized discrete -function.Comment: 25 pages; published version (IJMPC
Auditory power-law activation-avalanches exhibit a fundamental computational ground-state
The cochlea provides a biological information-processing paradigm that we
only begin to under- stand in its full complexity. Our work reveals an
interacting network of strongly nonlinear dynami- cal nodes, on which even
simple sound input triggers subnetworks of activated elements that follow
power-law size statistics ('avalanches'). From dynamical systems theory,
power-law size distribu- tions relate to a fundamental ground-state of
biological information processing. Learning destroys these power laws. These
results strongly modify the models of mammalian sound processing and provide a
novel methodological perspective for understanding how the brain processes
information.Comment: Videos are not included, please ask author
Virtual Evidence: A Constructive Semantics for Classical Logics
This article presents a computational semantics for classical logic using
constructive type theory. Such semantics seems impossible because classical
logic allows the Law of Excluded Middle (LEM), not accepted in constructive
logic since it does not have computational meaning. However, the apparently
oracular powers expressed in the LEM, that for any proposition P either it or
its negation, not P, is true can also be explained in terms of constructive
evidence that does not refer to "oracles for truth." Types with virtual
evidence and the constructive impossibility of negative evidence provide
sufficient semantic grounds for classical truth and have a simple computational
meaning. This idea is formalized using refinement types, a concept of
constructive type theory used since 1984 and explained here. A new axiom
creating virtual evidence fully retains the constructive meaning of the logical
operators in classical contexts.
Key Words: classical logic, constructive logic, intuitionistic logic,
propositions-as-types, constructive type theory, refinement types, double
negation translation, computational content, virtual evidenc
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