On the structure of graded transitive Lie algebras

We study finite-dimensional Lie algebras ${\mathfrak L}$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\bar{{\mathfrak g}}$, such that $\dfrac{\partial}{\partial x_i}\in \bar{{\mathfrak g}} \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\bar{{\mathfrak g}}$-module in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\bar{{\mathfrak g}}$ are described in detail, as well as all $\bar{{\mathfrak g}}$-modules that constitute such maximal ${\mathfrak L}$. All maximal ${\mathfrak L}$ are described explicitly for $n\leq 3$

On the structure of transitively differential algebras

We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain the vector fields ${\partial}/{\partial x_i} \; (i=1,\ldots, n)$ and $x_1{\partial}/{\partial x_1}+ \dots + x_n{\partial}/{\partial x_n}$. We derive some general results on the structure of such Lie algebras, and provide the complete classification in the cases $n=2$ and $n=3$. Finally we describe a certain construction in high dimensions

Symmetries of the WDVV equations and Chazy-type equations

We investigate the symmetry structure of the WDVV equations. We obtain an $r$-parameter group of symmetries, where $r = (n^2 + 7n + 2)/2 + \lfloor n/2 \rfloor$. Moreover it is proved that for $n=3$ and $n=4$ these comprise all symmetries. We determine a subgroup, which defines an $SL_2$-action on the space of solutions. For the special case $n=3$ this action is compared to the $SL_2$-symmetry of the Chazy equation. For $n=4$ and $n=5$ we construct new, Chazy-type, solutions

Polynomial solutions to the WDVV equations in four dimensions

All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately

Versal deformation of the Lie algebra L2L_2

We investigate deformations of the infinite dimensional vector field Lie algebra spanned by the fields $e_i = z^{i+1}d/dz$, where $i \ge 2$. The goal is to describe the base of a ``versal'' deformation; such a versal deformation induces all the other nonequivalent deformations and solves the deformation problem completely. \u

Clusterschemes in Dutch secondary schools

The first step in constructing timetables in secondary schools in Netherlands consists of constructing the clusterschemes for the higher classes. A clusterscheme contains clusterlines with optional subjects that will be taught in parallel; the problem is to divide these optional subjects in clusterlines, such that the number of hours needed is as low as possible. We describe an efficient branch-and-bound method for this problem. Moreover we describe a fast heuristic to assign students to subjectgroups. Some test results are presented

A convergent framework for the multicomponent KP-hierarchy

In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarchy, starting from a certain open subset of the Grassmann manifold of a special kind of Banach space, and derive an expression of its solutions in terms of Fredholm determinants. Further we show that the simplest nonscalar reduction of the present hierarchy leads to the AKNS-hierarchy

The norm of an averaging operator

We consider the operator $A: \ell^2 \to \ell^2$ defined by $A(a) = b$ for $a=(a_n)$ and $b=(b_n)$ with $b_n=\dfrac1n (a_1+a_2+\dots + a_n)$. We prove that $A$ has norm 2. \u

Eco-evolutionary dynamics on deformable fitness landscapes

Conventional approaches to modelling ecological dynamics often do not include evolutionary changes in the genetic makeup of component species and, conversely, conventional approaches to modelling evolutionary changes in the genetic makeup of a population often do not include ecological dynamics. But recently there has been considerable interest in understanding the interaction of evolutionary and ecological dynamics as coupled processes. However, in the context of complex multi-species ecosytems, especially where ecological and evolutionary timescales are similar, it is difficult to identify general organising principles that help us understand the structure and behaviour of complex ecosystems. Here we introduce a simple abstraction of coevolutionary interactions in a multi-species ecosystem. We model non-trophic ecological interactions based on a continuous but low-dimensional trait/niche space, where the location of each species in trait space affects the overlap of its resource utilisation with that of other species. The local depletion of available resources creates, in effect, a deformable fitness landscape that governs how the evolution of one species affects the selective pressures on other species. This enables us to study the coevolution of ecological interactions in an intuitive and easily visualisable manner. We observe that this model can exhibit either of the two behavioural modes discussed in the literature; namely, evolutionary stasis or Red Queen dynamics, i.e., continued evolutionary change. We find that which of these modes is observed depends on the lag or latency between the movement of a species in trait space and its effect on available resources. Specifically, if ecological change is nearly instantaneous compared to evolutionary change, stasis results; but conversely, if evolutionary timescales are closer to ecological timescales, such that resource depletion is not instantaneous on evolutionary timescales, then Red Queen dynamics result. We also observe that in the stasis mode, the overall utilisation of resources by the ecosystem is relatively efficient, with diverse species utilising different niches, whereas in the Red Queen mode the organisation of the ecosystem is such that species tend to clump together competing for overlapping resources. These models thereby suggest some basic conditions that influence the organisation of inter-species interactions and the balance of individual and collective adaptation in ecosystems, and likewise they also suggest factors that might be useful in engineering artificial coevolution