Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials dN on these dg algebras and conjecture that their homology matches that of the glN projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende

Crop rotations for grain production

There is an increasing demand for organically grown cereal grains in Denmark, which is expected to cause a change in the typical organic farm structure away from dairy farming and towards arable farming. Such a change may reduce the stability of the farming systems, because of decreasing soil fertility and problems with weed control. There have only been a limited number of studies under temperate conditions in Europe and North America, where different crop rotations have been compared under organic farming or similar production conditions

Learning from a long-term crop rotation experiment

A crop rotation experiment was established in 1996/97 at three locations representing differ-ent soil types and climates. Three factors were tested: i) crop rotation with different propor-tions of N2-fixing crops, ii) with and without a catch crop, and iii) with and without animal manure. A green manure crop increased yields in the following cereal crops, but at the rota-tional level, total yields were larger in crop rotations without a green manure crop. There were positive effects of animal manure and catch crops on yield. However, except for the coarse sandy soil, the yield effects of catch crops and animal manure decreased over time when a grass-clover green manure was included in the rotation. The problems with perennial weeds increased over time depending on crop rotation and use of catch crops. This stresses the im-portance of considering long-term effects in the evaluation of crop management measures

Design of an organic farming crop rotation experiment

A field experiment is conducted which focuses on crop rotations for cereal production in organic farming. The objective of the experiment is to explore the possibilities for both short-term and long-term increases in organic cereal production through manipulation of crop rotation design on different soil types. This paper describes the design of a rather complex experiment, and later papers will describe and discuss the results. Three factors are included in the experiment in a factorial design with two replicates: A) fraction of grass-clover and pulses in the rotation (crop rotation), B) catch crop (with or without catch crop or bi-cropped clover), and C) manure (with or without animal manure applied as slurry). All fields in all rotations are represented each year. The experimental factors are defined to allow management to be adjusted for optimisation of the individual treatment combinations. This makes the systems more realistic and the results more applicable in practical farming. The experiment is conducted at four locations representing major soil types and climate regions in Denmark. The main design criteria are related to requirements for a long-term experiment and the need of performing studies and experiments within the experiment itself

Crop production during the first course of an organic crop rotation trial in Denmark

Three factors are included in a factorial field experiment: 1) fraction of grass-clover and pulses in the rotation (crop rotation), 2) catch crop (with or without catch crop), and 3) manure (with or without animal manure). Manure is applied as slurry in rates corresponding to 40 % of the nitrogen demand. Grain yields for three sites and three years of the experiment are presented for two four-course rotations. One rotation has a green-manure crop, which is replaced by winter wheat in the other rotation. The use of manure significantly increased grain yield of the cereal crops in most cases. The positive effects of the catch crops were mainly observed in the spring cereals. The largest rotation yields were obtained in the crop rotation without a green manure crop. The positive effect of a green manure crop could not substitute for the yield decrease from leaving 25% of the area out of produc-tio

The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor

Nitrogen use efficiency of cereals in arable organic farming

The effect of nitrogen (N) supply and weeds on grain yield of spring barley, winter wheat and winter rye was investigated from 1997 to 2004 in an organic farming crop rotation experiment in Denmark on three soil types varying from coarse sand to sandy loam. Two experimental factors were included in the experiment in a factorial design: 1) catch crop (with and without), and 2) manure (with and without). The apparent recovery efficiency of N in grains (nitrogen use efficiency, NUE) from NH4-N in applied manure varied from 29 to 38% in spring barley and from 23 to 44% in winter cereals. The NUE of above-ground N in catch crops sampled in November prior to the spring barley varied from 16 to 52% with the largest value on the coarse sandy soil and the smallest value on the sandy loam soil. The NUE of N accumulated in grass-clover cuttings varied from 14 to 39% with the lowest value on the coarse sandy soil, most likely due to high rates of N leaching. The NUE declined with increasing amounts of N accumulated in the grass-clover cuttings. This indicates that grain yields can be improved by removing the grass-clover cuttings and applying the N contained in the cuttings in spring to the cereal crops, possibly after fermentation in a biogas reactor