Jordan cells in logarithmic limits of conformal field theory

It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of (unitary) minimal models in conformal field theory. Characters of quasi-rational representations are found to emerge as the limits of the associated irreducible Virasoro characters.Comment: 16 pages, v2: discussion of three-point functions and characters included; ref. added, v3: version to be publishe

Introduction of Organic Eprints

Organic Eprints is an open, on-line archive for research in organic food and farming with more than 10,000 publications - and growing rapidly. All use of the archive is free of charge. There are 15,000 registered users of Organic Eprints, and the archive has more than 175,000 visits each month. The archive contains scientific and popular articles, reports, presentations, project descriptions, books and other research publications. For each publication there is a short summary along with information about authors and contacts, publishing details, peer review status, subject area and research affiliation. In most cases, the full articles are freely available for download

Higher su(N) tensor products

We extend our recent results on ordinary su(N) tensor product multiplicities to higher su(N) tensor products. Particular emphasis is put on four-point couplings where the tensor product of four highest weight modules is considered. The number of times the singlet occurs in the decomposition is the associated multiplicity. In this framework, ordinary tensor products correspond to three-point couplings. As in that case, the four-point multiplicity may be expressed explicitly as a multiple sum measuring the discretised volume of a convex polytope. This description extends to higher-point couplings as well. We also address the problem of determining when a higher-point coupling exists, i.e., when the associated multiplicity is non-vanishing. The solution is a set of inequalities in the Dynkin labels.Comment: 17 pages, LaTe

Sowing time, false seedbed, row distance and mechanical weed control in organic winter wheat.

In organic farming, mechanical weed control in winter wheat is often difficult to carry out in the fall, and may damage the crop, and weed harrowing in the spring is not effective against erect, tap-rooted weeds such as Tripleurospermum inodorum, Papaver rhoeas, Brassica napus and others which have been established in the autumn. Some experiments concerning sowing strategy and intensity of mechanical weed control, which included row distance, were conducted. The results underline the importance of choosing weed control strategy, including preventive measures, according to the weed flora in the field. In the experiment with low weed pressure and without erect weeds, there was very little effect of sowing strategy and row distance. In such a case, the winter wheat might as well be sown early, in order to avoid possible yield loss by later sowing, and at normal row distance to enhance the competitiveness of the crop. In the experiments with high weed pressure and erect weeds, the weed control was better with late sowing and large row distance (high intensity control), even though this was not always reflected in the yield. However, the trade-off for lower input to the soil seed bank in organic systems should be enough to balance off the risk of smaller yield

Fusion multiplicities as polytope volumes: N-point and higher-genus su(2) fusion

We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N>3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide.Comment: 14 pages, LaTeX, version to be publishe

Polynomial Fusion Rings of Logarithmic Minimal Models

We identify quotient polynomial rings isomorphic to the recently found fundamental fusion algebras of logarithmic minimal models.Comment: 18 page

A non-reductive N=4 superconformal algebra

A new N=4 superconformal algebra (SCA) is presented. Its internal affine Lie algebra is based on the seven-dimensional Lie algebra su(2)\oplus g, where g should be identified with a four-dimensional non-reductive Lie algebra. Thus, it is the first known example of what we choose to call a non-reductive SCA. It contains a total of 16 generators and is obtained by a non-trivial In\"on\"u-Wigner contraction of the well-known large N=4 SCA. The recently discovered asymmetric N=4 SCA is a subalgebra of this new SCA. Finally, the possible affine extensions of the non-reductive Lie algebra g are classified. The two-form governing the extension appearing in the SCA differs from the ordinary Cartan-Killing form.Comment: 10 pages, LaTeX, version to be publishe

Fusion Algebras of Logarithmic Minimal Models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.Comment: 22 pages, v2: comments adde

Organic bread-wheat in New England, USA

In October 2010, researchers, farmers and millers from Maine and Vermont, USA, organized a trip to Denmark, in order to learn about local bread wheat production, milling and use from their more experienced counterparts with climates similar to their own. They have received a grant over four years for the project antitled Enhancing Farmers’ Capacity to Produce High Quality Organic Bread Wheat in which they will carry out research, development and education to improve the production and quality of organic bread wheat in the two states