Regenerative partition structures

We consider Kingman's partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures

Generalized Fleming-Viot processes with immigration via stochastic flows of partitions

The generalized Fleming-Viot processes were defined in 1999 by Donnelly and Kurtz using a particle model and by Bertoin and Le Gall in 2003 using stochastic flows of bridges. In both methods, the key argument used to characterize these processes is the duality between these processes and exchangeable coalescents. A larger class of coalescent processes, called distinguished coalescents, was set up recently to incorporate an immigration phenomenon in the underlying population. The purpose of this article is to define and characterize a class of probability-measure valued processes called the generalized Fleming-Viot processes with immigration. We consider some stochastic flows of partitions of Z_{+}, in the same spirit as Bertoin and Le Gall's flows, replacing roughly speaking, composition of bridges by coagulation of partitions. Identifying at any time a population with the integers $\mathbb{N}:=\{1,2,...\}$, the formalism of partitions is effective in the past as well as in the future especially when there are several simultaneous births. We show how a stochastic population may be directly embedded in the dual flow. An extra individual 0 will be viewed as an external generic immigrant ancestor, with a distinguished type, whose progeny represents the immigrants. The "modified" lookdown construction of Donnelly-Kurtz is recovered when no simultaneous multiple births nor immigration are taken into account. In the last part of the paper we give a sufficient criterion for the initial types extinction.Comment: typos and corrections in reference

Ontology and medical terminology: Why description logics are not enough

Ontology is currently perceived as the solution of first resort for all problems related to biomedical terminology, and the use of description logics is seen as a minimal requirement on adequate ontology-based systems. Contrary to common conceptions, however, description logics alone are not able to prevent incorrect representations; this is because they do not come with a theory indicating what is computed by using them, just as classical arithmetic does not tell us anything about the entities that are added or subtracted. In this paper we shall show that ontology is indeed an essential part of any solution to the problems of medical terminology – but only if it is understood in the right sort of way. Ontological engineering, we shall argue, should in every case go hand in hand with a sound ontological theory

Convolution operations arising from Vandermonde matrices

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.Comment: Submitted to IEEE Transactions on Information Theory. 16 pages, 1 figur

Making simple proofs simpler

An open partition \pi{} [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of \pi, the upset of B is a union of blocks of \pi. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root