9 research outputs found

### Constructing simply laced Lie algebras from extremal elements

For any finite graph Gamma and any field K of characteristic unequal to 2 we
construct an algebraic variety X over K whose K-points parameterise K-Lie
algebras generated by extremal elements, corresponding to the vertices of the
graph, with prescribed commutation relations, corresponding to the non-edges.
After that, we study the case where Gamma is a connected, simply laced Dynkin
diagram of finite or affine type. We prove that X is then an affine space, and
that all points in an open dense subset of X parameterise Lie algebras
isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this
fixed Lie algebra is the split finite-dimensional simple Lie algebra
corresponding to the associated finite-type Dynkin diagram. This gives a new
construction of these Lie algebras, in which they come together with
interesting degenerations, corresponding to points outside the open dense
subset. Our results may prove useful for recognising these Lie algebras.Comment: We made many corrections suggested by a referee, and extended our
results to positive characteristic greater than

### Extremal Presentations for Classical Lie Algebras

The long-root elements in Lie algebras of Chevalley type have been well
studied and can be characterized as extremal elements, that is, elements $x$
such that the image of (\ad x)^2 lies in the subspace spanned by $x$. In this
paper, assuming an algebraically closed base field of characteristic not 2, we
find presentations of the Lie algebras of classical Chevalley type by means of
minimal sets of extremal generators. The relations are described by simple
graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$,
and for $A_n$ the graph is the triangle connected to a path of length $n-3$.Comment: 26 pages, 6 figure

### Math saves the forest

Wireless sensor networks are decentralised networks consisting of sensors that can detect events and transmit data to neighbouring sensors. Ideally, this data is eventually gathered in a central base station. Wireless sensor networks have many possible applications. For example, they can be used to detect gas leaks in houses or fires in a forest.\ud
In this report, we study data gathering in wireless sensor networks with the objective of minimising the time to send event data to the base station. We focus on sensors with a limited cache and take into account both node and transmission failures. We present two cache strategies and analyse the performance of these strategies for specific networks. For the case without node failures we give the expected arrival time of event data at the base station for both a line and a 2D grid network. For the case with node failures we study the expected arrival time on two-dimensional networks through simulation, as well as the influence of the broadcast range

### A semi closed-form analytic pricing formula for call options in a hybrid Heston-Hull-White model

Without abstract

### Math saves the forest : Analysis and optimization of message delivery in wireless sensor networks

Wireless sensor networks are decentralised networks consisting of sensors that
can detect events and transmit data to neighbouring sensors. Ideally, this data is
eventually gathered in a central base station. Wireless sensor networks have many
possible applications. For example, they can be used to detect gas leaks in houses or
fires in a forest.
In this report, we study data gathering in wireless sensor networks with the objective
of minimising the time to send event data to the base station. We focus on
sensors with a limited cache and take into account both node and transmission failures.
We present two cache strategies and analyse the performance of these strategies
for specific networks. For the case without node failures we give the expected arrival
time of event data at the base station for both a line and a 2D grid network. For
the case with node failures we study the expected arrival time on two-dimensional
networks through simulation, as well as the influence of the broadcast range.
Keywords: sensor networks, data gathering, stochastic optimisation, distributed
algorithms, random walks, first-passage percolation