<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume8 Issue4" %>
Poisson Numbers and Poisson Distributions in Subset Surprisology
AndreasW.M. Dress1, T. Lokot2, L.D. Pustyl'nikov3, and W. Schubert4
1Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, 04103 Leipzig Germany
dress@mis.mpg.de
2Fakultät f¨šr Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
tlokot@mathematik.uni-bielefeld.de
3Keldysh Institute of Applied Mathematics of RAS, Miusskaja sq. 4, 125047, Moscow, Russia
pustylni@physik.uni-bielefeld.de
4Institute of Medical Neurobiology and MELTEC Ltd., University of Magdeburg 39120 Magdeburg, Germany
Schubert@pc.mdlink.de
Annals of Combinatorics 8 (4) p.473-485 December, 2004
AMS Subject Classification: 05A05, 05A10, 60C05, 62P10
Abstract:
Given a family of k+1 real-valued functions f0 , ..., fk defined on the set {1,..., n} and measuring the intensity of certain signals, we want to investigate whether these functions are 'dependent' or 'independent' by checking whether, for some given family of threshold values T0, ..., Tk, the size a of the collection of numbers whose signals exceed the corresponding threshold values T0, ..., Tk simultaneously for all 0, ..., k is surprisingly large (or small) in comparison to the family of cardinalities

of those numbers whose signals fi(j) individually exceed, for a given index i, the corresponding threshold value Ti. Such problems turn presently up in topological proteomics, a new direction of protein-interaction research that has become feasible due to new techniques developed in fluorescence microscopy called Multi-Epitope Ligand Cartography (or, for short, MELK = Multi-Epitop Liganden Kartographie). The above problem has led us to study the numbers of families of subsets with for and to investigate their asymptotic behaviour. In this note, we show that the associated probability distributions


defined on the set N0 of non-negative integers by


converge, with towards the Poisson distribution


for some fixed provided the numbers are assumed to converge with n to infinity in such a way that the conditions

are satisfied. Remarkably, it is the alternating signs in the expressions for resulting from the standard exclusion-inclusion principle that correspond to the alternating signs in the power series expression for when n turns to infinity.
Keywords: Subset surprisology, Poisson numbers, Poisson distribution, exclusion-inclusion principle, Möbius inversion, topological proteomics, Multi-Epitope Ligand Cartography, Multi-Epitop Liganden Kartographie, MELK

References:

1. W. Feller, An Introduction to Probability Theory and Its Applications. vol. 1, third edition, New York, 1968.

2. A. Dress, T. Lokot, and L.D. Pustyl'nikov, Poisson Law in Subset Surprisology, Proceedings of the ZiFWorkshop "The Science of Complexity, Mathematics to Technology to a Sustainable World". N2001/105,
http://www.physik.uni-bielefeld.de/complexity/pub- lications .html.

3. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge, University Press, 1992.

4. W. Schubert, Topological proteomics, toponomics, MELK-technology, In: Proteomics of Microorganismus: Fundamental Aspects and Application, Advances in Biochemical Engeneering/ Biotechnology, Vol. 83, M. vHecker, S. M¨šllner, Eds., Springer Verlag, Berlin - Heidelberg - New York, 2003.

5. V.A. Vatutin and V.G. Mikhailov, Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles, Theory Prob. Appl. 27 (1982) 734--743.

6. V.F. Kolchin, B.A. Sevast'yanov, and V.P. Chistyakov, Random Allocation, Jon Wiley, New York-Toronto-London, 1978.