Baddeley a et al stochastic geometry CIME lecs martina franca 2004 (LNM 1892 2007)(ISBN 3540381740)(301s)

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1892 Fondazione C.I.M.E Firenze C.I.M.E means Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Center Conceived in the early fifties, it was born in 1954 and made welcome by the world mathematical community where it remains in good health and spirit Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities during the past years So they already know what the C.I.M.E is all about For the benefit of future potential users and co-operators the main purposes and the functioning of the Centre may be summarized as follows: every year, during the summer, Sessions (three or four as a rule) on different themes from pure and applied mathematics are offered by application to mathematicians from all countries Each session is generally based on three or four main courses (24−30 hours over a period of 6-8 working days) held from specialists of international renown, plus a certain number of seminars A C.I.M.E Session, therefore, is neither a Symposium, nor just a School, but maybe a blend of both The aim is that of bringing to the attention of younger researchers the origins, later developments, and perspectives of some branch of live mathematics The topics of the courses are generally of international resonance and the participation of the courses cover the expertise of different countries and continents Such combination, gave an excellent opportunity to young participants to be acquainted with the most advance research in the topics of the courses and the possibility of an interchange with the world famous specialists The full immersion atmosphere of the courses and the daily exchange among participants are a first building brick in the edifice of international collaboration in mathematical research C.I.M.E Director Pietro ZECCA Dipartimento di Energetica “S Stecco” Università di Firenze Via S Marta, 50139 Florence Italy e-mail: zecca@unifi.it C.I.M.E Secretary Elvira MASCOLO Dipartimento di Matematica Università di Firenze viale G.B Morgagni 67/A 50134 Florence Italy e-mail: mascolo@math.unifi.it For more information see CIME’s homepage: http://www.cime.unifi.it CIME’s activity is supported by: – Istituto Nationale di Alta Matematica “F Severi” – Ministero dell’Istruzione, dell’Università e della Ricerca – Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione, Ufficio V A Baddeley · I Bárány R Schneider · W Weil Stochastic Geometry Lectures given at the C.I.M.E Summer School held in Martina Franca, Italy, September 13–18, 2004 With additional contributions by D Hug, V Capasso, E Villa Editor: W Weil ABC Authors, Editor and Contributors Rolf Schneider Daniel Hug Adrian Baddeley School of Mathematics & Statistics University of Western Australia Nedlands WA 6009 Australia e-mail: adrian@maths.uwa.edu.au Mathematisches Institut Albert-Ludwigs-Universität Eckerstr 79104 Freiburg i Br Germany e-mail: rolf.schneider@math.uni-freiburg.de daniel.hug@math.uni-freiburg.de Imre Bárány Rényi Institute of Mathematics 1364 Budapest Pf 127 Hungary e-mail: barany@renyi.hu Wolfgang Weil Mathematisches Institut II Universität Karlsruhe 76128 Karlsruhe Germany e-mail: weil@math.uni-karlsruhe.de and Mathematics University College London Gower Street London, WC1E 6BT United Kingdom Vincenzo Capasso Elena Villa Department of Mathematics University of Milan via Saldini 50 20133 Milano Italy e-mail: vincenzo.capasso@mat.unimi.it villa@mat.unimi.it Library of Congress Control Number: 2006931679 Mathematics Subject Classification (2000): Primary 60D05 Secondary 60G55, 62H11, 52A22, 53C65 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-38174-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-38174-7 Springer Berlin Heidelberg New York DOI 10.1007/3-540-38174-0 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting by the authors and SPi using a Springer LATEX package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper SPIN: 11815334 41/SPi 543210 Preface The mathematical treatment of random geometric structures can be traced back to the 18th century (the Buffon needle problem) Subsequent considerations led to the two disciplines Integral Geometry and Geometric Probability, which are connected with the names of Crofton, Herglotz, Blaschke (to mention only a few) and culminated in the book of Santal´ o (Integral Geometry and Geometric Probability, 1976) Around this time (the early seventies), the necessity grew to have new and more flexible models for the description of random patterns in Biology, Medicine and Image Analysis A theory of Random Sets was developed independently by D.G Kendall and Matheron In connection with Integral Geometry and the already existing theory of Point Processes the new field Stochastic Geometry was born Its rapid development was influenced by applications in Spatial Statistics and Stereology Whereas at the beginning emphasis was laid on models based on stationary and isotropic Poisson processes, the recent years showed results of increasing generality, for nonisotropic or even inhomogeneous structures and without the strong independence properties of the Poisson distribution On the one side, these recent developments in Stochastic Geometry went hand-in-hand with a fresh interest in Integral Geometry, namely local formulae for curvature measures (in the spirit of Federer’s Geometric Measure Theory) On the other side, new models of point processes (Gibbs processes, Strauss processes, hardcore and cluster processes) and their effective simulation (Markov Chain Monte Carlo, perfect simulation) tightened the close relation between Stochastic Geometry and Spatial Statistics A further, very interesting direction is the investigation of spatial-temporal processes (tumor growth, communication networks, crystallization processes) The demand for random geometric models is steadily growing in almost all natural sciences or technical fields The intention of the Summer School was to present an up-to-date description of important parts of Stochastic Geometry The course took place in Martina Franca from Monday, September 13, to Friday, September 18, 2004 It was attended by 49 participants (including the lecturers) The main lecturers were Adrian Baddeley (University of Western Australia, Perth), Imre VI Preface B´ar´ any (University College, London, and Hungarian Academy of Sciences, Budapest), Rolf Schneider (University of Freiburg, Germany) and Wolfgang Weil (University of Karlsruhe, Germany) Each of them gave four lectures of 90 minutes which we shortly describe, in the following Adrian Baddeley spoke on Spatial Point Processes and their Applications He started with an introduction to point processes and marked point processes in Rd as models for spatial data and described the basic notions (counting measures, intensity, finite-dimensional distributions, capacity functional) He explained the construction of the basic model in spatial statistics, the Poisson process (on general locally compact spaces), and its transformations (thinning and clustering) He then discussed higher order moment measures and related concepts (K function, pair correlation function) In his third lecture, he discussed conditioning of point processes (conditional intensity, Palm distributions) and the important Campbell-Mecke theorem The Palm distributions lead to G and J functions which are of simple form for Poisson processes In the last lecture he considered point processes in bounded regions and described methods to fit corresponding models to given data He illustrated his lectures by computer simulations Imre B´ar´ any spoke on Random Points, Convex Bodies, and Approximation He considered the asymptotic behavior of functionals like volume, number of vertices, number of facets, etc of random convex polytopes arising as convex hulls of n i.i.d random points in a convex body K ⊂ Rd Starting with a short historical introduction (Efron’s identity, formulas of R´enyi and Sulanke), he emphasized the different limit behavior of expected functionals for smooth bodies K on one side and for polytopes K on the other side In order to explain this difference, he showed that the asymptotic behavior of the expected missed volume E(K, n) of the random convex hull behaves asymptotically like the volume of a deterministic set, namely the shell between K and the cap body of K (the floating body) This result uses Macbeath regions and the ‘economic cap covering theorem’ as main tools The results were extended to the expected number of vertices and of facets In the third lecture, random approximation (approximation of K by the convex hull of random points) was compared with best approximation (approximation from inside w.r.t minimal missed volume) It was shown that random approximation is almost as good as best approximation A further comparison concerned convex hulls of lattice points in K In the last lecture, for a convex body K ⊂ R2 , the probability p(n, K) that n random points in K are in convex position was considered and the asymptotic behavior (as n → ∞) was given (extension of the classical Sylvester problem) The lectures of Rolf Schneider concentrated on Integral Geometric Tools for Stochastic Geometry In the first lecture, the classical results from integral geometry, the principal kinematic formulas and the Crofton formulas were given in their general form, for intrinsic volumes of convex bodies (which were introduced by means of the Steiner formula) Then, Hadwiger’s characterization theorem for additive functionals was explained and used to Preface VII generalize the integral formulas In the second lecture, local versions of the integral formulas for support measures (curvature measures) and extensions to sets in the convex ring were discussed This included a local Steiner formula for convex bodies Extensions to arbitrary closed sets were mentioned The third lecture presented translative integral formulas, in local and global versions, and their iterations The occurring mixed measures and functionals were discussed in more detail and connections to support functions and mixed volumes were outlined The last lecture studied general notions of k-dimensional area and general Crofton formulas Relations between hyperplane measures and generalized zonoids were given It was shown how such relations can be used in stochastic geometry, for example, to give estimates for the intersection intensity of a general (non-stationary) Poisson hyperplane process in Rd Wolfgang Weil, in his lectures on Random Sets (in Particular Boolean Models), built upon the previous lectures of A Baddeley and R Schneider He first gave an introduction to random closed sets and particle processes (point processes of compact sets, marked point processes) and introduced the basic model in stochastic geometry, the Boolean model (the union set of a Poisson particle process) He described the decomposition of the intensity measure of a stationary particle process and used this to introduce the two quantities which characterize a Boolean model (intensity and grain distribution) He also explained the role of the capacity functional (Choquet’s theorem) and its explicit form for Boolean models which shows the relation to Steiner’s formula In the second lecture, mean value formulas for additive functionals were discussed They lead to the notion of density (quermass density, density of area measure, etc.) which was studied then for general random closed sets and particle processes The principal kinematic and translative formulas were used to obtain explicit formulas for quermass densities of stationary and isotropic Boolean models as well as for non-isotropic Boolean models (with convex or polyconvex grains) in Rd Statistical consequences were discussed for d = and d = and ergodic properties were shortly mentioned The third lecture was concerned with extensions in various directions: densities for directional data and their relation to associated convex bodies (with an application to the mean visible volume of a Boolean model), interpretation of densities as Radon-Nikodym derivatives of associated random measures, density formulas for non-stationary Boolean models In the final lecture, random closed sets and Boolean models were investigated from outside by means of contact distributions Recent extensions of this concept were discussed (generalized directed contact distributions) and it was explained that in some cases they suffice to determine the grain distribution of a Boolean model completely The role of convexity for explicit formulas of contact distributions was discussed and, as the final result, it was explained that the polynomial behavior of the logarithmic linear contact distribution of a stationary and isotropic Boolean model characterizes convexity of the grains Since the four lecture series could only cover some parts of stochastic geometry, two additional lectures of 90 minutes were included in the program, VIII Preface given by D Hug and V Capasso Daniel Hug (University of Freiburg) spoke on Random Mosaics as special particle processes He presented formulas for the different intensities (number and content of faces) for general mosaics and for Voronoi mosaics and then explained a recent solution to Kendall’s conjecture concerning the asymptotic shape of large cells in a Poisson Voronoi mosaic Vincenzo Capasso (University of Milano) spoke on Crystallization Processes as spatio-temporal extensions of point processes and Boolean models and emphasized some problems arising from applications The participants presented themselves in some short contributions, at one afternoon, as well as in two evening sessions The attendance of the lectures was extraordinarily good Most of the participants had already some background in spatial statistics or stochastic geometry Nevertheless, the lectures presented during the week provided the audience with a lot of new material for subsequent studies These lecture notes contain (partially extended) versions of the four main courses (and the two additional lectures) and are also intended as an information of a wider readership about this important field I thank all the authors for their careful preparation of the manuscripts I also take the opportunity, on behalf of all participants, to thank C.I.M.E for the effective organization of this summer school; in particular, I want to thank Vincenzo Capasso who initiated the idea of a workshop on stochastic geometry Finally, we were all quite grateful for the kind hospitality of the city of Martina Franca Karlsruhe, August 2005 Wolfgang Weil Contents Spatial Point Processes and their Applications Adrian Baddeley Point Processes 1.1 Point Processes in 1D and 2D 1.2 Formulation of Point Processes 1.3 Example: Binomial Process 1.4 Foundations 1.5 Poisson Processes 1.6 Distributional Characterisation 1.7 Transforming a Point Process 1.8 Marked Point Processes 1.9 Distances in Point Processes 1.10 Estimation from Data 1.11 Computer Exercises Moments and Summary Statistics 2.1 Intensity 2.2 Intensity for Marked Point Processes 2.3 Second Moment Measures 2.4 Second Moments for Stationary Processes 2.5 The K-function 2.6 Estimation from Data 2.7 Exercises Conditioning 3.1 Motivation 3.2 Palm Distribution 3.3 Palm Distribution for Stationary Processes 3.4 Nearest Neighbour Function 3.5 Conditional Intensity 3.6 J-function 3.7 Exercises Modelling and Statistical Inference 2 12 16 19 21 23 24 26 26 30 32 35 38 39 40 42 42 44 49 51 52 55 56 57 X Contents 4.1 Motivation 4.2 Parametric Modelling and Inference 4.3 Finite Point Processes 4.4 Point Process Densities 4.5 Conditional Intensity 4.6 Finite Gibbs Models 4.7 Parameter Estimation 4.8 Estimating Equations 4.9 Likelihood Devices References 57 58 61 62 64 66 69 70 72 73 Random Polytopes, Convex Bodies, and Approximation Imre B´ ar´ any 77 Introduction 77 Computing Eφ(Kn ) 79 Minimal Caps and a General Result 80 The Volume of the Wet Part 82 The Economic Cap Covering Theorem 84 Macbeath Regions 84 Proofs of the Properties of the M -regions 87 Proof of the Cap Covering Theorem 89 Auxiliary Lemmas from Probability 92 10 Proof of Theorem 3.1 95 11 Proof of Theorem 4.1 96 12 Proof of (4) 98 13 Expectation of fk (Kn ) 101 14 Proof of Lemma 13.2 102 15 Further Results 104 16 Lattice Polytopes 108 17 Approximation 109 18 How It All Began: Segments on the Surface of K 114 References 115 Integral Geometric Tools for Stochastic Geometry Rolf Schneider 119 Introduction 119 From Hitting Probabilities to Kinematic Formulae 120 1.1 A Heuristic Question on Hitting Probabilities 120 1.2 Steiner Formula and Intrinsic Volumes 123 1.3 Hadwiger’s Characterization Theorem for Intrinsic Volumes 126 1.4 Integral Geometric Formulae 129 Localizations and Extensions 136 2.1 The Kinematic Formula for Curvature Measures 136 2.2 Additive Extension to Polyconvex Sets 143 2.3 Curvature Measures for More General Sets 148 Mean Densities for Inhomogeneous Birth-and-growth Processes 277 fx (t) = px (t)h(t, x), from which we immediately obtain ∂VV (t, x) = (1 − VV (t, x))h(t, x) ∂t (8) This is an extension of the well known Avrami-Kolmogorov formula [18, 3], proven for space homogeneous birth and growth rates; our expression holds whenever a mean volume density and an hazard function are well defined In order to obtain explicit evolution equations for the relevant densities of the birth-and-growth process, we need to make explicit their dependence upon the kinetic parameters of the processes of birth (such as the intensity of the corresponding marked point process) and growth (the growth rate) This can be done by means of the hazard function h(t, x) In general cases (e.g not Poissonian cases) the expressions for the survival and the hazard functions are quite complicated, because they must take into account all the previous history of the process (cf (9)) So, it can be useful to estimate the hazard function, directly For example, under our assumptions, the following holds (see [13]) h(t, x) = G(t, x) ∂ HS,Θt (r, x)|r=0 , ∂r where HS,Θt is the local spherical contact distribution function of Θt For the birth-and-growth model defined above, we may introduce a causal cone associated with a point x ∈ E and a time t > 0, as defined by Kolmogorov [18], A(t, x) := {(y, s) ∈ E × [0, t] | x ∈ Θts (y)} We denote by Cs (t, x) the section of the causal cone at time s < t, Cs (t, x) := {y ∈ E | (y, s) ∈ A(t, x)} = {y ∈ E | x ∈ Θts (y)} It is possible to obtain (see [13]) expressions for the survival and the hazard functions in terms of the birth process N and of the causal cone For example, under our modelling assumptions, if N has continuous intensity measure, then for any fixed t t px (t) = exp − E(˜ ν (ds)k(s, Cs (t, x)) | N [A(s, Cs (t, x))] = 0) (9) In the Poisson case, the independence property of increments makes this expression simpler 278 Vincenzo Capasso and Elena Villa space x3 x x1 A(t,x) x2 t1 t2 t3 t time Fig The causal cone A(t, x) of point x at time t: it is the space-time region where a nucleation has to take place so that point x is reached by a growing grain by time t The Poisson Case If N is a marked Poisson process, it is easily seen that px (t) = P(x ∈ Θt ) = P(N (A(t, x)) = 0) = e−ν0 (A(t,x)) , where ν0 (A(t, x)) is the volume of the causal cone with respect to the free space intensity of the Poisson process: ν0 (A(t, x)) = α(s, y)dsdy A(t,x) So, we have ∂ ∂ ln px (t) = ν0 (A(t, x)) ∂t ∂t The following theorem holds [8] whose proof requires sophisticate techniques of PDE’s in a geometric measure theoretic setting h(t, x) = − Theorem 4.2 Under our modelling assumptions on birth and on growth, which make the evolution problem well posed, the following equality holds ∂ ν0 (A(t, x)) = G(t, x) ∂t t Rd K(t0 , x0 ; t, x)α(t0 , x0 )dx0 dt0 (10) with K(t0 , x0 ; t, x) := {z∈Rd |τ (t0 ,x0 ;z)=t} δ(z − x)da(z) Here δ is the Dirac function, da(z) is a (d − 1)-dimensional surface element, and τ (t0 , x0 ; z) is the solution of the eikonal problem ∂τ ∂τ (t0 , x0 , x) = (t0 , x0 , x), ∂x0 G(t0 , x0 ) ∂t0 Mean Densities for Inhomogeneous Birth-and-growth Processes 279 ∂τ (t0 , x0 , x) = ∂x G(τ (t0 , x0 , x), x) In our case it has been shown [12] that the volume of the causal cone can be expressed in terms of the extended volume density Theorem 4.3 Under the previous modelling assumptions on birth and on growth, the following equality holds ν0 (A(t, x)) = Vex (t, x) (11) As a consequence h(t, x) = ∂ ∂ ν0 (A(t, x)) = Vex (t, x) ∂t ∂t so that ∂ ∂ VV (t, x) = (1 − VV (t, x)) Vex (t, x) ∂t ∂t This equation is exactly the Kolmogorov-Avrami formula extended to a birthand-growth process with space-time inhomogeneous parameters of birth and of growth [18, 3] Finally, by direct comparison between (11), (10) and (7), we may claim that t Sex (t, x) = Rd K(t0 , x0 ; t, x)α(t0 , x0 )dx0 dt0 and h(t, x) = G(t, x)Sex (t, x) Consequently, by remembering (6) and (8), we obtain SV (t, x) = (1 − VV (t, x))Sex (t, x) The availability of an explicit expression for the hazard function, in terms of the relevant kinetic parameters of the process, makes it possible to obtain evolution equations for all facet densities of the Johnson-Mehl tessellation associated with our birth-and-growth process (see [17],[22],[11]) A major problem arises when the birth-and-growth process is strongly coupled with an underlying field (such as temperature in a crystallization process); in such a case the kinetic parameters of the stochastic birth-andgrowth process are themselves stochastic, depending on the history of the crystallization process As a consequence the previous theory for obtaining evolution equations for densities cannot be applied The interested reader may refer to [10] and [8], for an approximate solution of the problem by means of hybrid models that take into account the possible multiple scale structure of the problem Acknowledgements It is a pleasure to acknowledge the contribution of M Burger in Linz and A Micheletti in the development of joint research projects relevant for this presentation 280 Vincenzo Capasso and Elena Villa References Araujo, A., Gin´e, E.: The Central Limit Theorem for Real and Banach Valued Random Variables John Wiley & Sons, New York (1980) Ash, R.B.: Real Analysis and Probability Academic Press, New York (1972) Avrami A.: Kinetic of phase change Part I J Chem Phys., 7, 1103–112 (1939) Baddeley, A.J., Molchanov, I.S.: On the expected measure of a random set In: Jeulin, D (ed) Advances in Theory and Applications of Random Sets, Proc International Symposium, Oct 9-11, 1996, Fontainebleau, France World Sci Publ., Singapore (1997) Barles, G., Soner, H.M., Souganidis, P.E.: Front propagation and phase-field theory SIAM J Control Optim., 31, 439–469 (1993) Br´emaud, P.: Point Processes and Queues, Martingale Dynamics Springer, New York (1981) Burger, M.: Growth fronts of first-order Hamilton-Jacobi equations SFB Report 02-8, J Kepler University, Linz (2002) Burger, M., Capasso, V., Pizzocchero, L.: Mesoscale averaging of nucleation and growth models Submitted Burger, M., Capasso, V., Salani, C.: Modelling multi-dimensional crystallization of polymers in interaction with heat transfer Nonlinear Anal Real World Appl., 3, 139–160 (2002) 10 Capasso, V (ed): Mathematical Modelling for Polymer Processing Polymerization, Crystallization, Manufacturing Mathematics in Industry Series, 2, Springer, Heidelberg (2003) 11 Capasso, V., Micheletti, A.: Stochastic geometry of spatially structured birthand-growth processes Application to crystallization processes In: Merzbach, E (ed) Topics in Spatial Stochastic Processes Lecture Notes in Mathematics (CIME Subseries), 1802, Springer, Heidelberg (2002) 12 Capasso, V., Micheletti, A.: On the hazard function for inhomogeneous birthand-growth processes Submitted 13 Capasso, V., Villa, E.: Survival functions and contact distribution functions for inhomogeneous, stochastic geometric marked point processes Stoch Anal Appl., 23, 79–96 (2005) 14 Capasso, V., Villa, E.: On the stochastic geometric densities of time dependent random closed sets In preparation 15 Federer, H.: Geometric Measure Theory Springer, Berlin (1996) 16 Hug, D., Last, G., Weil, W.: A local Steiner-type formula for general closed sets and applications Math Z., 246, 237–272 (2004) 17 Johnson, W.A., Mehl, R.F.: Reaction Kinetics in processes of nucleation and growth Trans A.I.M.M.E., 135, 416–458 (1939) 18 Kolmogorov, A.N.: On the statistical theory of the crystallization of metals Bull Acad Sci USSR, Math Ser., 1, 355–359 (1937) 19 Kolmogorov, A.N.: Foundations of the Theory of Probability Chelsea Pub Co., New York (1956) 20 Last, G., Brandt, A.: Marked Point Processes on the Real Line The Dynamic Approach Springer, New York (1995) 21 Lorenz, T.: Set valued maps for image segmentation Comput Visual Sci., 4, 41–57 (2001) 22 Møller, J.: Random Johnson-Mehl tessellations Adv in Appl Probab., 24, 814–844 (1992) Mean Densities for Inhomogeneous Birth-and-growth Processes 281 23 Morale, D.: A stochastic particle model for vasculogenesis: a multiple scale approach In: Capasso, V (ed.) Mathematical Modelling and Computing in Biology and Medicine The MIRIAM Project Series, ESCULAPIO Pub Co., Bologna (2003) 24 Serini, G., et al.: Modeling the early stages of vascular network assembly EMBO J., 22, 1771–1779 (2003) 25 Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization Shape Sensitivity Analysis Springer, Berlin (1992) 26 Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Application John Wiley & Sons, New York (1995) 27 Ubukata, T.: Computer modelling of microscopic features of molluscan shells In: Sekimura, T., et al (eds) Morphogenesis and Pattern Formation in Biological Systems Springer, Tokyo (2003) 28 Ză ahle, M.: Random processes of Hausdor rectiable closed sets Math Nachr., 108, 49–72 (1982) List of Participants Aletti Giacomo University of Milano, Italy giacomo.aletti@unimi.it Azarina Svetlana Voronezh State University, Russia azarinas@mail.ru Baddeley Adrian University of Western Australia, Australia adrian@maths.uwa.edu.au (lecturer) B´ ar´ any Imre R´ enyi Institute of Mathematics Budapest, Hungary barany@renyi.hu (lecturer) Berchtold Maik Swiss Federal Institute of Techn., Switzerland berchtld@stat.math.ethz.ch Bianchi Annamaria University of Milano, Italy abianchi@mat.unimi.it Bianchi Gabriele University of Florence, Italy gabriele.bianchi@unifi.it Bobyleva Olga Moscow State University, Russia o bobyleva@mail.ru Bongiorno Enea University of Milano, Italy bongiorno.enea@tiscali.it 10 Capasso Vincenzo University of Milano, Italy Vincenzo.Capasso@unimi.it 11 Cerd` an Ana Universidad de Alicante, Spain aacs@alu.ua.es 12 Cerny Rostislav Charles University Prague, Czech Republic rostislav.cerny@karlin.mff.cuni.cz 13 Connor Stephen University of Warwick, UK s.b.connor@warwick.ac.uk 14 Fleischer Frank University of Ulm, Germany ffrank@mathematik.uni-ulm.de 15 Gallois David France Telecom R&D david.gallois@rd.francetelecom.com 16 Gille Wilfried University of Halle, Germany gille@physik.uni-halle.de 17 Gots Ekaterina Voronezh State University, Russia kgots@aport2000.ru 18 Hoffmann Lars Michael University of Karlsruhe, Germany LarsHoffmann@ePost.de 19 Hug Daniel University of Freiburg, Germany daniel.hug@math.uni-freiburg.de ´ 20 J´ onsd´ ottir Kristjana Yr Aarhus University, Denmark kyj@imf.au.dk 21 Karamzin Dmitry Computing Centre RAS, Russia dmitry karamzin@mail.ru 22 Kozlova Ekaterina Moscow State University Lomonosov, Russia ekozlova@fors.ru 284 List of Participants 23 Lautensack Claudia Inst Techno- und Wirtschaftsmathematik Kaiserslautern, Germany lautensack@itwm.fraunhofer.de 24 Legland David INRA, France david.legland@jouy.inra.fr 25 Lhotsky Jiri Charles University Prague, Czech Republic jiri.lhotsky@email.cz 26 Mannini Claudio University of Florence, Italy claudio.mannini@dicea.unifi.it 27 Micheletti Alessandra University of Milano, Italy alessandra.micheletti@unimi.it 28 Miori Cinzia Universidad de Alicante, Spain cm4@alu.ua.es 29 Morale Daniela University of Milano, Italy morale@mat.unimi.it 30 Nazin Sergey Institute of Control Sciences RAS, Russia snazin@ipu.rssi.ru 31 Ortisi Matteo University of Milano, Italy ortisi@mat.unimi.it 32 Pantle Ursa University of Ulm, Germany pantle@mathematik.uni-ulm.de 33 Salani Paolo University of Florence, Italy salani@math.unifi.it 34 Sapozhnikov Artyom Heriot-Watt University, UK artyom@ma.hw.ac.uk 35 Scarsini Marco University of Torino, Italy marco.scarsini@unito.it 36 Schmaehling Jochen University of Heidelberg, Germany jochen.schmaehling@de.bosch.com 37 Schmidt Hendrik University of Ulm, Germany hendrik@mathematik.uni-ulm.de 38 Schneider Rolf University of Freiburg, Germany rolf.schneider@math.uni-freiburg.de (lecturer) 39 Schuhmacher Dominic University of Zurich, Switzerland schumi@amath.unizh.ch 40 Shcherbakov Vadim CWI, Netherlands V.Shcherbakov@cwi.nl 41 Sicco Alessandro University of Torino, Italy sicco@dm.unito.it 42 Sirovich Roberta University of Torino, Italy sirovich@dm.unito.it 43 Solanes Gil University of Stuttgart, Germany solanes@mathematik.uni-stuttgart.de 44 Thorarinsdottir Thordis Linda University of Aarhus, Denmark disa@imf.au.dk 45 Tontchev Nikolay University of Berne, Switzerland nito@stat.unibe.ch 46 Villa Elena University of Milano, Italy villa@mat.unimi.it 47 Voss Christian University of Rostock, Germany christian.voss@mathematik.unirostock.de 48 Weil Wolfgang University of Karlsruhe, Germany weil@math.uni-karlsruhe.de (lecturer, editor) 49 Winter Steffen University of Jena, Germany winter@minet.uni-jena.de LIST OF C.I.M.E SEMINARS Published by C.I.M.E 1954 Analisi funzionale Quadratura delle superficie e questioni connesse Equazioni differenziali non lineari 1955 Teorema di Riemann-Roch e questioni connesse Teoria dei numeri Topologia Teorie non linearizzate in elasticit` a, idrodinamica, aerodinamic Geometria proiettivo-differenziale 1956 Equazioni alle derivate parziali a caratteristiche reali 10 Propagazione delle onde elettromagnetiche automorfe 11 Teoria della funzioni di pi` u variabili complesse e delle funzioni 1957 12 Geometria aritmetica e algebrica (2 vol.) 13 Integrali singolari e questioni connesse 14 Teoria della turbolenza (2 vol.) 1958 15 Vedute e problemi attuali in relativit` a generale 16 Problemi di geometria differenziale in grande 17 Il principio di minimo e le sue applicazioni alle equazioni funzionali 1959 18 Induzione e statistica 19 Teoria algebrica dei meccanismi automatici (2 vol.) 20 Gruppi, anelli di Lie e teoria della coomologia 1960 21 Sistemi dinamici e teoremi ergodici 22 Forme differenziali e loro integrali 1961 23 Geometria del calcolo delle variazioni (2 vol.) 24 Teoria delle distribuzioni 25 Onde superficiali 1962 26 Topologia differenziale 27 Autovalori e autosoluzioni 28 Magnetofluidodinamica 1963 29 Equazioni differenziali astratte 30 Funzioni e variet` a complesse 31 Propriet` a di media e teoremi di confronto in Fisica Matematica 1964 32 33 34 35 1965 36 Non-linear continuum theories 37 Some aspects of ring theory 38 Mathematical optimization in economics Relativit` a generale Dinamica dei gas rarefatti Alcune questioni di analisi numerica Equazioni differenziali non lineari Published by Ed Cremonese, Firenze 1966 39 40 41 42 Calculus of variations Economia matematica Classi caratteristiche e questioni connesse Some aspects of diffusion theory 1967 43 Modern questions of celestial mechanics 44 Numerical analysis of partial differential equations 45 Geometry of homogeneous bounded domains 1968 46 Controllability and observability 47 Pseudo-differential operators 48 Aspects of mathematical logic 1969 49 Potential theory 50 Non-linear continuum theories in mechanics and physics and their applications 51 Questions of algebraic varieties 1970 52 53 54 55 1971 56 Stereodynamics 57 Constructive aspects of functional analysis (2 vol.) 58 Categories and commutative algebra 1972 59 Non-linear mechanics 60 Finite geometric structures and their applications 61 Geometric measure theory and minimal 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to appear 164 SPDE in Hydrodynamics: Recent Progress and Prospects to appear 2006 165 Pseudo-Differential Operators, Quantization and Signals 166 Mixed Finite Elements, Compatibility Conditions, and Applications 167 From a Microscopic to a Macroscopic Description of Complex Systems 168 Quantum Transport: Modelling, Analysis and Asymptotics Stochastic Methods in Finance Hyperbolic Systems of Balance Laws Symplectic 4-Manifolds and Algebraic Surfaces Mathematical Foundation of Turbulent Viscous Flows 1740) 1784) 1823) 1715) 1734) (LNM 1848) (LNM 1891) to appear (LNM 1856) to appear to appear (LNM 1871) to appear to appear (LNM 1892) announced announced announced announced Lecture Notes in Mathematics For information about earlier volumes please contact your bookseller or Springer LNM Online archive: springerlink.com Vol 1701: Ti-Jun Xiao, J Liang, The Cauchy Problem of Higher Order Abstract Differential Equations (1998) Vol 1702: J Ma, J Yong, Forward-Backward Stochastic Differential 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Spatial Point Processes and their Applications Adrian Baddeley School of Mathematics & Statistics, University of Western Australia Nedlands WA 6009, Australia e-mail: adrian@maths.uwa.edu.au A. .. 38 Adrian Baddeley 2.5 The K-function Second moment properties are important in the statistical analysis of spatial point pattern data, just as the sample variance is important in classical statistics... disease (‘spatial epidemiology’ [19]) Spatial point processes also serve as a basic model in random set theory [42] and image analysis [41] For general surveys of applications of spatial point processes,