4,430 research outputs found
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical
statements, based on computability with Ordinal Turing Machines (OTMs), one of
which resembles Turing reducibility while the other is modelled after Weihrauch
reducibility. We give sample applications by showing that certain (algebraic)
constructions are not effective in the OTM-sense and considerung the effective
equivalence of various versions of the axiom of choice
The gauge structure of generalised diffeomorphisms
We investigate the generalised diffeomorphisms in M-theory, which are gauge
transformations unifying diffeomorphisms and tensor gauge transformations.
After giving an En(n)-covariant description of the gauge transformations and
their commutators, we show that the gauge algebra is infinitely reducible,
i.e., the tower of ghosts for ghosts is infinite. The Jacobiator of generalised
diffeomorphisms gives such a reducibility transformation. We give a concrete
description of the ghost structure, and demonstrate that the infinite sums give
the correct (regularised) number of degrees of freedom. The ghost towers belong
to the sequences of rep- resentations previously observed appearing in tensor
hierarchies and Borcherds algebras. All calculations rely on the section
condition, which we reformulate as a linear condition on the cotangent
directions. The analysis holds for n < 8. At n = 8, where the dual gravity
field becomes relevant, the natural guess for the gauge parameter and its
reducibility still yields the correct counting of gauge parameters.Comment: 24 pp., plain tex, 1 figure. v2: minor changes, including a few added
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Quantization of Even-Dimensional Actions of Chern-Simons Form with Infinite Reducibility
We investigate the quantization of even-dimensional topological actions of
Chern-Simons form which were proposed previously. We quantize the actions by
Lagrangian and Hamiltonian formulations {\`a} la Batalin, Fradkin and
Vilkovisky. The models turn out to be infinitely reducible and thus we need
infinite number of ghosts and antighosts. The minimal actions of Lagrangian
formulation which satisfy the master equation of Batalin and Vilkovisky have
the same Chern-Simons form as the starting classical actions. In the
Hamiltonian formulation we have used the formulation of cohomological
perturbation and explicitly shown that the gauge-fixed actions of both
formulations coincide even though the classical action breaks Dirac's
regularity condition. We find an interesting relation that the BRST charge of
Hamiltonian formulation is the odd-dimensional fermionic counterpart of the
topological action of Chern-Simons form. Although the quantization of two
dimensional models which include both bosonic and fermionic gauge fields are
investigated in detail, it is straightforward to extend the quantization into
arbitrary even dimensions. This completes the quantization of previously
proposed topological gravities in two and four dimensions.Comment: 50 pages, latex, no figure
Shapes From Pixels
Continuous-domain visual signals are usually captured as discrete (digital)
images. This operation is not invertible in general, in the sense that the
continuous-domain signal cannot be exactly reconstructed based on the discrete
image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness).
In this paper, we study the problem of recovering shape images with smooth
boundaries from a set of samples. Thus, the reconstructed image is constrained
to regenerate the same samples (consistency), as well as forming a shape
(bilevel) image. We initially formulate the reconstruction technique by
minimizing the shape perimeter over the set of consistent binary shapes. Next,
we relax the non-convex shape constraint to transform the problem into
minimizing the total variation over consistent non-negative-valued images. We
also introduce a requirement (called reducibility) that guarantees equivalence
between the two problems. We illustrate that the reducibility property
effectively sets a requirement on the minimum sampling density. One can draw
analogy between the reducibility property and the so-called restricted isometry
property (RIP) in compressed sensing which establishes the equivalence of the
minimization with the relaxed minimization. We also evaluate
the performance of the relaxed alternative in various numerical experiments.Comment: 13 pages, 14 figure
Covariant theory of asymptotic symmetries, conservation laws and central charges
Under suitable assumptions on the boundary conditions, it is shown that there
is a bijective correspondence between equivalence classes of asymptotic
reducibility parameters and asymptotically conserved n-2 forms in the context
of Lagrangian gauge theories. The asymptotic reducibility parameters can be
interpreted as asymptotic Killing vector fields of the background, with
asymptotic behaviour determined by a new dynamical condition. A universal
formula for asymptotically conserved n-2 forms in terms of the reducibility
parameters is derived. Sufficient conditions for finiteness of the charges
built out of the asymptotically conserved n-2 forms and for the existence of a
Lie algebra g among equivalence classes of asymptotic reducibility parameters
are given. The representation of g in terms of the charges may be centrally
extended. An explicit and covariant formula for the central charges is
constructed. They are shown to be 2-cocycles on the Lie algebra g. The general
considerations and formulas are applied to electrodynamics, Yang-Mills theory
and Einstein gravity.Comment: 86 pages Latex file; minor correction
Quasi-periodically driven quantum systems
Floquet theory provides rigorous foundations for the theory of periodically
driven quantum systems. In the case of non-periodic driving, however, the
situation is not so well understood. Here, we provide a critical review of the
theoretical framework developed for quasi-periodically driven quantum systems.
Although the theoretical footing is still under development, we argue that
quasi-periodically driven quantum systems can be treated with generalizations
of Floquet theory in suitable parameter regimes. Moreover, we provide a
generalization of the Floquet-Magnus expansion and argue that quasi-periodic
driving offers a promising route for quantum simulations
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