22,353 research outputs found
Some Empirical Criteria for Attributing Creativity to a Computer Program
Peer reviewedPostprin
Branes And Supergroups
Extending previous work that involved D3-branes ending on a fivebrane with
, we consider a similar two-sided problem. This
construction, in case the fivebrane is of NS type, is associated to the
three-dimensional Chern-Simons theory of a supergroup U or OSp
rather than an ordinary Lie group as in the one-sided case. By -duality, we
deduce a dual magnetic description of the supergroup Chern-Simons theory; a
slightly different duality, in the orthosymplectic case, leads to a strong-weak
coupling duality between certain supergroup Chern-Simons theories on
; and a further -duality leads to a version of Khovanov
homology for supergroups. Some cases of these statements are known in the
literature. We analyze how these dualities act on line and surface operators.Comment: 143 page
Statistical Understanding of Quark and Lepton Masses in Gaussian Landscapes
The fundamental theory of nature may allow a large landscape of vacua. Even
if the theory contains a unified gauge symmetry, the 22 flavor parameters of
the Standard Model, including neutrino masses, may be largely determined by the
statistics of this landscape, and not by any symmetry. Then the measured values
of the flavor parameters do not lead to any fundamental symmetries, but are
statistical accidents; their precise values do not provide any insights into
the fundamental theory, rather the overall pattern of flavor reflects the
underlying landscape. We investigate whether random selection from the
statistics of a simple landscape can explain the broad patterns of quark,
charged lepton, and neutrino masses and mixings. We propose Gaussian landscapes
as simplified models of landscapes where Yukawa couplings result from overlap
integrals of zero-mode wavefunctions in higher-dimensional supersymmetric gauge
theories. In terms of just five free parameters, such landscapes can account
for all gross features of flavor, including: the hierarchy of quark and charged
lepton masses; small quark mixing angles, with 13 mixing less than 12 and 23
mixing; very light Majorana neutrino masses, with the solar to atmospheric
neutrino mass ratio consistent with data; distributions for leptonic 12 and 23
mixings that are peaked at large values, while the distribution for 13 mixing
is peaked at low values; and order unity CP violating phases in both the quark
and lepton sectors. While the statistical distributions for flavor parameters
are broad, the distributions are robust to changes in the geometry of the extra
dimensions. Constraining the distributions by loose cuts about observed values
leads to narrower distributions for neutrino measurements of 13 mixing, CP
violation, and neutrinoless double beta decay.Comment: 86 pages, 26 figures, 2 tables, and table of content
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
Effective Complexity and its Relation to Logical Depth
Effective complexity measures the information content of the regularities of
an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of
the disadvantages of Kolmogorov complexity, also known as algorithmic
information content. In this paper, we give a precise formal definition of
effective complexity and rigorous proofs of its basic properties. In
particular, we show that incompressible binary strings are effectively simple,
and we prove the existence of strings that have effective complexity close to
their lengths. Furthermore, we show that effective complexity is related to
Bennett's logical depth: If the effective complexity of a string exceeds a
certain explicit threshold then that string must have astronomically large
depth; otherwise, the depth can be arbitrarily small.Comment: 14 pages, 2 figure
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