Table Of ContentMathEverywhere
G. Aletti M. Burger A. Micheletti D. Morale
Editors
Math Everywhere
Deterministic and Stochastic Modelling
in Biomedicine, Economics and Industry.
Dedicated to
the 60th Birthday of Vincenzo Capasso
ABC
GiacomoAletti MartinBurger
AlessandraMicheletti InstitutfürNumerischeund
DanielaMorale AngewandteMathematik
WestfälischeWilhelms-UniversitätMünster
ADAMSS&DepartmentofMathematics
Einsteinstraße62
UniversityofMilano
48149Münster,Germany
ViaC.Saldini50
E-mail:[email protected]
20133Milano,Italy
E-mail:[email protected]
[email protected]
[email protected]
Mathematics Subject Classification (2000): 35Kxx, 60D05, 28A75, 62M30, 62G05,
60G17,60H10,65C30,92B05
LibraryofCongressControlNumber:2006933717
ISBN-10 3-540-44445-9SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-44445-9SpringerBerlinHeidelbergNewYork
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“Do you really want to
mathematize everything?”
Jacques-Louis Lions
Preface
The title of the book is a perfect way to synthesize the point of view of Vin-
cenzo Capasso (VK) about the role played by mathematics in nature and all
human activities, as mentioned by Willi Ja¨ger in the Introduction. As VK
mentioned in one of his speeches, “it is important that our scientific com-
munity appreciates the importance of theory and mathematics to face the
enormouschallengeswhicharisefromtheemergingfieldsofBiologyandMedi-
cine, Industry and Economics to improve the quality of Life.” His enthusiasm
and new ideas have brought him in this challenging mission. By transferring
methods and tools from one field to another, by learning novel mathemat-
ical methods and ideas from the observation of real world phenomena and
behavior, and by approaching real industrial problems, he has accepted to
play a fundamental role in both the Italian and European community in the
establishment and reinforcement of communication channels between Acad-
emia (methods) and Industry (applications), being himself a driving force of
the scientific and technological progress that we are observing in our daily
life.
This books collects the proceedings of the workshop Math Everywhere:
Deterministic and Stochastic Modelling in Biomedicine, Economics and In-
dustry which took place in Milano, Italy in the period September 4–6, 2005,
and was organized in honor of the 60th birthday of Vincenzo Capasso (VK).
It has been a successful meeting involving many prestigious scientists, in par-
ticular mathematicians from all over the world who have discussed, in a very
friendly and relaxed atmosphere, relevant innovative mathematical methods
which have been developed to solve mathematically complex systems arisen
in connection with real industrial, economic and social problems. The main
featureofthemeetinghasbeenthevarietyofthethemesdiscussedbothfrom
themathematicalandtheappliedpointofview,atanimpressivelyhighscien-
tificlevel.IthasreflectedthewidespectrumofscientificinterestsofVincenzo
Capasso, which have been ranging from the first papers on the mathematical
foundations of Quantum Mechanics in the early 70s, through his fundamen-
tal work on reaction-diffusion systems, including the qualitative behaviour,
VIII Preface
control problems and stochasticity of epidemic systems, published in many
papers and a very cited book “Mathematical structures of epidemic systems”
(an editorial success of the Lecture Notes in Biomathematics, published by
Springer), up to the most recent work on time-and-space structured systems
with evolving stochastic geometries. In this work, in which he has involved a
numberofstudents/collaboratorsfromvariousuniversitiesinEurope,aseries
of open problems have been faced including the strong coupling of evolu-
tion equations for the (stochastic) geometries with the evolution equations
of the underlying fields, at different scales. He has opened new mathematical
problems at the interface of geometric measure theory, stochastic geometry,
stochastic analysis, and statistics for evolution problems regarding densities,
concentrations, i.e. relevant quantities describing the phenomena from an ap-
plied point of view, so to match one of the most important goals in VK’s
research, being useful to the applied sciences.
Alongtheselines,theoreticallyorientedpapershavebeenpresentedinthe
field of reaction-diffusion systems, control theory, stochastic geometry, multi-
parameter stochastic processes and stochastic interacting particles.
TheworkshophasbeenamirrorofVK’slivelyexperience:frombeingthe
PresidentofESMTB,theEuropeanSocietyforMathematicalandTheoretical
BiologyandthePresidentofECMI,theEuropeanConsortiumofMathematics
in Industry; by establishing at first MIRIAM, the Milan Research Centre
for Industrial and Applied Mathematics, and more recently ADAMSS, the
InterdisciplinaryCentreforMathematicalModelling,StatisticalAnalysis,and
ComputationalSimulationforScientificandTechnologicalInnovation,aiming
at a daily interaction between Mathematics and all other research areas, to
further innovation in Industry, Finance, Medicine, public services, etc., in
collaboration with a number of other centres, universities and public research
bodies,companies,organicallylinkedwithsimilaractivitiesatcommonlevels
of excellence in Europe, and elsewhere.
HererecentworksinBiomathematicsandMathematicalMedicinearepre-
sented: Structured Populations and interacting Individuals, Tumour Cords,
Vascular Cancer, Vasculogenesis, Epidemics Models, Competition Models,
Glucose-Insulin Homeostasis, Pattern Formation in Butterflies, Quiescent
States,ReactionDiffusionSystem,andNeutronLifetimeEstimationaresome
of the main subjects investigated nowadays in the field. Furthermore in the
industrial applications problems on the kinetics of nucleation and growth
processes, crystallization phenomena and optimizing batch processes in the
chemicalindustryhavebeeninvestigated.Finallyalsotheproblemofoptimal
marketing in economics is presented here.
It is worth to mention here that the opening lecture and the closing lec-
ture have been delivered by Grace Yang and Willi Ja¨ger, respectively. Grace
Yang has always been quoted by VK as his most influential scientific advisor,
especially for introducing himself scientifically in one of the most prestigious
SchoolsofMathematicalStatistics,theoneleadbyJerzyNeymanandLucien
LeCam, whom, thanks to her, he had the chance to meet and discuss with.
Preface IX
Indeed his first relevant paper in probability was on the the asymptotics of
the Neyman-Scott model of epidemics, a pioneering model of stochastic point
processes;itwaswrittenbyanenthusiasticandbrilliantyoungscientistunder
the guidance of Grace, first during the Summer School on Biomathematics at
the Scuola Normale Superiore in Pisa (1972), and later at the Department of
Mathematics of the University of Maryland (1973). Willi Ja¨ger has not only
strongly influenced VK scientifically, by stimulating and encouraging him to
face hard mathematical problems in the general area of nonlinear multiscale
systems,butasWilliJa¨gerhasreported,theyhavedeeplysharedsuchaview
of Mathematics Everywhere, supported by a genuine interest in facing chal-
lenging problems in biomedicine of great relevance for the quality of life of
humanbeings,bymeansofdifferentmathematicalmethodsandoftencreating
problem-driven new ones.
With elation we are happy to state that everyone has very much enjoyed
the very warm atmosphere, and the fantastic richness of scientific ideas that
have pervaded the whole meeting.
X Preface
Here we would like to thank VK, for sharing with us his enthusiasm and
thisideaofamathematicsasatoolforachallengingmission.Yourenthusiasm
and happiness of being a mathematician, but first of all a human being have
soaked in our being scientists. Thanks!
Milano, Linz, Giacomo Aletti
July 2006 Martin Burger
Alessandra Micheletti
Daniela Morale
Contents
List of Contributors ...........................................XV
Introduction
Willi Ja¨ger...................................................... 1
Part I Deterministic and Stochastic Systems
Coupled Dynamics and Quiescent Phases
Karl P. Hadeler, Thomas Hillen ................................... 7
Long Time Behavior of a System of Stochastic Differential
Equations Modelling Aggregation
Vincenzo Capasso, Daniela Morale, Matteo Ortisi .................... 25
Invariant Density Estimation for Multidimensional Diffusions
Annamaria Bianchi .............................................. 39
First Contact Distribution Function Estimation for a Partially
Observed Dynamic Germ-Grain Model with Renewal
Dropping Process
Marcello De Giosa ............................................... 51
An Extension of the Kolmogorov-Avrami Formula to
Inhomogeneous Birth-and-Growth Processes
Martin Burger, Vincenzo Capasso, Alessandra Micheletti.............. 63
On the Generalized Geometric Densities of Random Closed
Sets. An Application to Growth Processes
Vincenzo Capasso, Elena Villa .................................... 77
The Multiparameter Fractional Brownian Motion
Erick Herbin, Ely Merzbach ....................................... 93
XII Contents
Filtering of Multiparameter Processes: Theory and
Applications
Alberto Carabar´ın-Aguirre, B. Gail Ivanoff, Adriana Jordan ..........103
IFSM Representation of Brownian Motion with Applications
to Simulation
Stefano Maria Iacus, Davide La Torre ..............................115
Iterated Function Systems on Multifunctions
Davide La Torre, Franklin Mendivil, Edward R. Vrscay ...............125
Part II Mathematical Problems in Biology, Medicine and Ecology
Stochastic Modeling and Estimation in a Neutron Lifetime
Experiment
Grace L. Yang, Kevin J. Coakley...................................141
The Transport of Specific Monoclonal Antibodies in Tumour
Cords
Alessandro Bertuzzi, Antonio Fasano, Alberto Gandolfi, Carmela
Sinisgalli........................................................151
Structural Adaptation in Normal and Cancerous Vasculature
Philip K. Maini, Toma´s Alarco´n, Helen M. Byrne, Markus R. Owen,
James Murphy...................................................165
Approximation of 2D and 3D Models of Chemotactic Cell
Movement in Vasculogenesis
Fausto Cavalli, Andrea Gamba, Giovanni Naldi, Matteo Semplice ......179
Homogenization Closure For A Two-Dimensional Effective
Model Describing Fluid-Structure Interaction in Blood Flow
Andro Mikeli´c, Sunˇcica Cˇani´c .....................................193
Pattern Formation in Butterfly Wings: Experiments and
Models
Toshio Sekimura .................................................207
Stabilization for a Reaction-Diffusion System in Epidemiology
Sebastian Ani¸ta..................................................219
Global Stability of Equilibria for a Metapopulation S–I–S
Model
Francesca Arrigoni, Andrea Pugliese ...............................229
State Feedback Control of the Glucose-Insulin System
Pasquale Palumbo, Andrea De Gaetano .............................241
