Global branching laws by global Okounkov bodies

Let $G'$ be a complex semisimple group, and let $G \subseteq G'$ be a semisimple subgroup. We show that the branching cone of the pair $(G, G')$, which (asymptotically) parametrizes all pairs $(W, V)$ of irreducible finite-dimensional $G$-representations $W$ which occur as subrepresentations of a finite-dimensional irreducible $G'$-representation $V$, can be identified with the pseudo-effective cone, \overline{\mbox{Eff}}(Y), of some GIT quotient $Y$ of the flag variety of the group $G \times G'$. Moreover, we prove that the quotient $Y$ is a Mori dream space. As a consequence, the global Okounkov body $\Delta(Y)$ of $Y$, with respect to some admissible flag of subvarieties of $Y$, is fibred over the branching cone of $(G, G')$, and the fibre $\Delta(Y)_{(W, V)}$ over a point $(W, V)$ carries information about (the asymptotics of) the multiplicity of $W$ in $V$. Using the global Okounkov body $\Delta(Y)$, we easily derive a multi-dimensional generalization of Okounkov's result about the log-concavity of asymptotic multiplicities

Reusability in software engineering

This paper surveys recent work concerning reusability in software engineering. The current directions in software reusability are discussed, and the two major approaches of reusable building blocks and reusable patterns studied. An extensive bibliography, parts of which are annotated, is included

Okounkov bodies for ample line bundles

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq...\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope

Activity theoretical view on crop rotation planning in organic vegetable farming

Crop rotation is an essential basis of an organic farming system. An activity theoretical concept of object shall be used in examining planning processes of crop rotations. Farmers' object construction means their creating and maintaining the social meaning and purpose of the material farming activity. I shall assume the farmers´ object construction in planning crop rotations to reflect their overall object in organic vegetable farming. First, this paper examines theoretically the object in organic vegetable farming by devising a framework of different types of object constructions. Second, two farms with organic vegetable production will be described, and the farmers´ objects will be shown in the light of the histories of the farms. This will show that the different types of objects have not evolved at random. Third, the dynamic movement of the object construction in the crop rotation planning processes is explored. The results will show that the farmers´ object, although historically understood, is not fixed. On the contrary, the object is in a constant move and even contradictory

Farmers' joint reflection on strategies: example of using a learning tool

The presentation of different farms was essential for using the learning tool. The way farmers modified the framework show locally important strategic dimensions. Strategic decisions go beyond the farm gate. Farmers' community seems to be important for motivation

Okounkov bodies for ample line bundles with applications to multiplicities for group representations

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope. In the case when $X$ is a homogeneous surface, and the pseudoeffective cone of $X$ is rational polyhedral, we also show that the global Okounkov body is a rational polyhedral cone if the flag of subvarieties is suitably chosen. Finally, we provide an application to the asymptotic study of group representations.Comment: supersedes the preprint arXiv:1007.191

Discussion - cooperation in local organic food systems

What is needed for local food systems to emerge? First, time. Second, active people are needed. In order to be sustainable local and ecological food chains need to be, at least to some extent, economically profitable

Introduction

The purpose of this publication is to describe the cases around the Baltic Sea that are involved or linked to the BERAS project. In them, active people, projects and organizations have taken initiatives towards local and organic food chains and cooperation

How to find and understand development problems and learning challenges in organic vegetable farming?

Laura Seppänen's doctoral dissertation in agroecology, using an approach derived from cultural historical activity theory, examines the developmental problems and learning challenges in organic vegetable farming

Global Okounkov bodies for Bott-Samelson varieties

We use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the counting of lattice points in polytopes.Comment: A new and simpler definition of a good flag is introduced, and Bott-Samelson varieties are shown to admit such flag