213,070 research outputs found
A decidable subclass of finitary programs
Answer set programming - the most popular problem solving paradigm based on
logic programs - has been recently extended to support uninterpreted function
symbols. All of these approaches have some limitation. In this paper we propose
a class of programs called FP2 that enjoys a different trade-off between
expressiveness and complexity. FP2 programs enjoy the following unique
combination of properties: (i) the ability of expressing predicates with
infinite extensions; (ii) full support for predicates with arbitrary arity;
(iii) decidability of FP2 membership checking; (iv) decidability of skeptical
and credulous stable model reasoning for call-safe queries. Odd cycles are
supported by composing FP2 programs with argument restricted programs
Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks
Attractors in asymmetric neural networks with deterministic parallel dynamics
were shown to present a "chaotic" regime at symmetry eta < 0.5, where the
average length of the cycles increases exponentially with system size, and an
oscillatory regime at high symmetry, where the typical length of the cycles is
2. We show, both with analytic arguments and numerically, that there is a sharp
transition, at a critical symmetry \e_c=0.33, between a phase where the
typical cycles have length 2 and basins of attraction of vanishing weight and a
phase where the typical cycles are exponentially long with system size, and the
weights of their attraction basins are distributed as in a Random Map with
reversal symmetry. The time-scale after which cycles are reached grows
exponentially with system size , and the exponent vanishes in the symmetric
limit, where . The transition can be related to the dynamics
of the infinite system (where cycles are never reached), using the closing
probabilities as a tool.
We also study the relaxation of the function ,
where is the local field experienced by the neuron . In the symmetric
system, it plays the role of a Ljapunov function which drives the system
towards its minima through steepest descent. This interpretation survives, even
if only on the average, also for small asymmetry. This acts like an effective
temperature: the larger is the asymmetry, the faster is the relaxation of ,
and the higher is the asymptotic value reached. reachs very deep minima in
the fixed points of the dynamics, which are reached with vanishing probability,
and attains a larger value on the typical attractors, which are cycles of
length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge
Predictable trajectories of the reduced Collatz iteration and a possible pathway to the proof of the Collatz conjecture (Version 2)
I show here that there are three different kinds of iterations for the
reduced Collatz algorithm; depending on whether the root of the number is odd
or even. There is only one kind of iteration if the root is odd and two kinds
if the root is even. I also show that iterations on numbers with odd roots will
cause an increase in value and eventually lead to an even rooted number. The
iterations on even rooted numbers will subsequently cause a decrease in values.
Because increase in values during the odd root iterations are bounded, I
conclude that the Collatz iteration cannot veer to infinity. Since the sequence
generated by the Collatz iteration is infinite and the values of the numbers do
not veer to infinity it must either cycle and/or converge. I postulate that any
cycling must occur with alternating types of iterations: e.g. odd rooted
iterations which cause the values of the numbers to increase followed by even
rooted iterations which causes the values to decrease. I show here that for
simpler types of cycles, valid values of odd rooted or even rooted numbers are
only found in a narrow gap which closes as the number of iterations increase. I
further generalize to all types of odd-even and even-odd iterations. Given that
previous work has shown that only very large non-trivial cycles are feasible
during the Collatz iteration and this study shows the low probability of large
simple cycles, leads us to the conclusion most likely cycles other than the
trivial cycle are not possible during the Collatz iteration
Algebraic cycles and the classical groups II: Quaternionic cycles
In part I of this work we studied the spaces of real algebraic cycles on a
complex projective space P(V), where V carries a real structure, and completely
determined their homotopy type. We also extended some functors in K-theory to
algebraic cycles, establishing a direct relationship to characteristic classes
for the classical groups, specially Stiefel-Whitney classes. In this sequel, we
establish corresponding results in the case where V has a quaternionic
structure. The determination of the homotopy type of quaternionic algebraic
cycles is more involved than in the real case, but has a similarly simple
description. The stabilized space of quaternionic algebraic cycles admits a
nontrivial infinite loop space structure yielding, in particular, a delooping
of the total Pontrjagin class map. This stabilized space is directly related to
an extended notion of quaternionic spaces and bundles (KH-theory), in analogy
with Atiyah's real spaces and KR-theory, and the characteristic classes that we
introduce for these objects are nontrivial. The paper ends with various
examples and applications.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper27.abs.htm
Differentials in the homological homotopy fixed point spectral sequence
We analyze in homological terms the homotopy fixed point spectrum of a
T-equivariant commutative S-algebra R. There is a homological homotopy fixed
point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)),
converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of
the homotopy fixed point spectrum. We show that there are Dyer-Lashof
operations beta^epsilon Q^i acting on this algebra spectral sequence, and that
its differentials are completely determined by those originating on the
vertical axis. More surprisingly, we show that for each class x in the
$^{2r}-term of the spectral sequence there are 2r other classes in the
E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite
cycles, i.e., survive to the E^infty-term. We apply this to completely
determine the differentials in the homological homotopy fixed point spectral
sequences for the topological Hochschild homology spectra R = THH(B) of many
S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all
finite subgroups C of T, and for the Tate- and homotopy orbit spectral
sequences. This work is part of a homological approach to calculating
topological cyclic homology and algebraic K-theory of commutative S-algebras.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-27.abs.htm
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