1,489 research outputs found
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
Tine options for alleviating compaction in wheelings
Repeated trafficking and harvesting operations lead to high levels of compaction in inter-row wheelings used in asparagus (Asparagus officinalis) production. This reduces soil porosity and infiltration resulting in water ponding on the soil surface. Even on gently sloping land this can result in runoff generation and an increased risk of soil erosion. A winged tine (WT) is currently used by a leading asparagus grower to loosen compacted inter-row wheelings. In order to test the effectiveness of this tine for alleviating compaction and implications for runoff and soil erosion control, it was evaluated alongside several other tine configurations. These were a narrow tine (NT); a narrow tine with two shallow leading tines (NSLT); a winged tine with two shallow leading tines (WSLT); and a modified para-plough (MPP). Testing was conducted under controlled conditions on a sandy loam soil in the Soil Management Facility at Cranfield University, Bedfordshire, UK. Tine performance was assessed at 3 depths (175, 250 and 300 mm) by draught force; soil disturbance (both above and below ground); specific draught for a given level of soil disturbance; surface roughness; and estimated change in soil bulk density. The effectiveness of tines for compaction alleviation and potential for mitigating runoff and soil erosion varied with depth. The most effective tines were found to be the MPP NSLT and the WSLT at 175 mm, 250 mm and 300 mm depth, respectively
The density of critical percolation clusters touching the boundaries of strips and squares
We consider the density of two-dimensional critical percolation clusters,
constrained to touch one or both boundaries, in infinite strips, half-infinite
strips, and squares, as well as several related quantities for the infinite
strip. Our theoretical results follow from conformal field theory, and are
compared with high-precision numerical simulation. For example, we show that
the density of clusters touching both boundaries of an infinite strip of unit
width (i.e. crossing clusters) is proportional to .
We also determine numerically contours for the density of clusters crossing
squares and long rectangles with open boundaries on the sides, and compare with
theory for the density along an edge.Comment: 11 pages, 6 figures. Minor revision
Factorization of correlations in two-dimensional percolation on the plane and torus
Recently, Delfino and Viti have examined the factorization of the three-point
density correlation function P_3 at the percolation point in terms of the
two-point density correlation functions P_2. According to conformal invariance,
this factorization is exact on the infinite plane, such that the ratio R(z_1,
z_2, z_3) = P_3(z_1, z_2, z_3) [P_2(z_1, z_2) P_2(z_1, z_3) P_2(z_2,
z_3)]^{1/2} is not only universal but also a constant, independent of the z_i,
and in fact an operator product expansion (OPE) coefficient. Delfino and Viti
analytically calculate its value (1.022013...) for percolation, in agreement
with the numerical value 1.022 found previously in a study of R on the
conformally equivalent cylinder. In this paper we confirm the factorization on
the plane numerically using periodic lattices (tori) of very large size, which
locally approximate a plane. We also investigate the general behavior of R on
the torus, and find a minimum value of R approx. 1.0132 when the three points
are maximally separated. In addition, we present a simplified expression for R
on the plane as a function of the SLE parameter kappa.Comment: Small corrections (final version). In press, J. Phys.
Anchored Critical Percolation Clusters and 2-D Electrostatics
We consider the densities of clusters, at the percolation point of a
two-dimensional system, which are anchored in various ways to an edge. These
quantities are calculated by use of conformal field theory and computer
simulations. We find that they are given by simple functions of the potentials
of 2-D electrostatic dipoles, and that a kind of superposition {\it cum}
factorization applies. Our results broaden this connection, already known from
previous studies, and we present evidence that it is more generally valid. An
exact result similar to the Kirkwood superposition approximation emerges.Comment: 4 pages, 1 (color) figure. More numerics, minor corrections,
references adde
Percolation Crossing Formulas and Conformal Field Theory
Using conformal field theory, we derive several new crossing formulas at the
two-dimensional percolation point. High-precision simulation confirms these
results. Integrating them gives a unified derivation of Cardy's formula for the
horizontal crossing probability , Watts' formula for the
horizontal-vertical crossing probability , and Cardy's formula for
the expected number of clusters crossing horizontally . The
main step in our approach implies the identification of the derivative of one
primary operator with another. We present operator identities that support this
idea and suggest the presence of additional symmetry in conformal field
theories.Comment: 12 pages, 5 figures. Numerics improved; minor correction
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