Analysis of non-equilibrium evolution problems:
selected topics in material and life sciences

Common challenges exist in both material sciences and life sciences in regard to obtaining and understanding macroscopic descriptions of phenomena. The multitude of examples is vast and can be found in a variety of places (molecular and biophysical chemistry, reaction kinetics and material science, as well as in evolutionary, molecular biology and ecology...)

The challenges are generally twofold: On the one hand one wishes to find accurate description of the microscale, while on the other hand, having in view certain microscale dynamics (close to micro phase transitions), one wishes the basic understanding over much larger scales. The second challenge is the one we wish to touch with this workshop.

This meeting wants to bring together Dutch mathematicians working in the analysis of Partial Differential Equations (PDEs) with direct applications to material and life sciences together with overseas colleagues, working in the same field, stemming from fife distinct schools of Partial Differential Equations.

Regardless whether we consider a piece of metal or concrete material, or a biological tissue, or a group of agents, the main keyword of the meeting is “Localizing and Understanding Non-Equilibrium Phase Transitions”.

Specifically we will address analytic aspects of nonlinear systems of PDEs arising particularly in modeling the macroscopic behavior of materials such as shape-memory alloys, porous materials (e.g. concrete), biological membranes, or chemotactic motion of agglomerated populations.

The discussions will be not only around typical analysis topics like well-posedness, specific limits (e.g. fast-reaction asymptotics, large-time behavior, homogenization), nonlinear micro/macro transmission conditions, coupling mechanics equations with thermal and chemistry models, but will also include the analysis of hybrid systems - where discrete (microscopic) interactions, capable to emerge well-defined macroscopic patterns, are coupled with phenomenological macroscopic descriptions.

This meeting wants to be the continuation of the Lorentz Center workshop “PDE Approximations in Fast Reaction-Slow Diffusion Scenarios” (10-14/11/2008, organized by T. Aiki, D. Hilhorst, M. Mimura, and A. Muntean), of the more recent workshop: “4th Euro-Japanese Workshop on Blow Up” (Lorentz Center , 6-11/09/2010, organized by J. Hulshof et al.) as well as of two special sections on PDEs in Material Sciences (Arlington, USA, 2008) and (Dresden, Germany, 2010) under the umbrella of the AIMS (American Institute of Mathematical Sciences) – DCDS (Discrete and Continuous Dynamical Systems).

The journal – DCDS Series S (Discrete and Continuous Dynamical Systems) has agreed to publish the best original contributions presented in our workshop.

09.00 - 10.00	Mitsuharu Otani	On Brinkman-Forchheimer equations of flow in double-diffusive convection
10.10 - 11.10	Georg Prokert	Traveling Waves in a Hele-Shaw Type Moving Boundary Problem
11.10 - 11.40	Coffee/tea
11.40 - 12.40	Tetsuya Ishiwata	Behavior of polygonal curves by crystalline curvature flow
12.40 - 13.40	Lunch
13.40 - 14.10	Yabin Fan	Travelling wave solution for some degenerate pseudo-parabolic equations
14.10 - 14.40	Risei Kano	The existence of solutions for tumor invasion models equipped with a diffusion depending on some stress
14.40 - 15.40	Jens Rademacher	Coherent structures and patterns in Magnetization
15.40 - 16.00	Coffee/tea
16.00 - 16.30	Hiroshi Watanabe	Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients
16.30 - 17.00	Toyohiko Aiki	Large-time behavior of a free boundaru problem describing concrete carbonation
17.00 - 17.30	Okihiro Sawada	On the norm-inflation solutions to the Navier-Stokes equations in the critical space
17.30 - 18.00	Robert Nolet	Existence of solutions for the diffusive VSC model

09.00 - 10.00	Iuliu Sorin Pop	The mathematics of salt: from pore to core
10.10 - 11.10	Goto Akagi	Nonlinear diffusion equations involving p(x)-Laplacians
11.10 - 11.40	Coffee/tea
11.40 - 12.40	Nobuyuki Kenmochi	Revival Models of Human and Economic Activities in Disaster Areas
12.40 - 12.50	Posterprize Ceremony and Closing

A mathematical model for a concrete carbonation problem has been proposed by Böhm and Muntean in 2006. Also, they showed some mathematical results on their model. In the model a carbonation front is given as a curve s(t) for time variable t > 0. In order to improve their results we have simplified their model and obtained theorems on existence and uniqueness of a solution to the model. Recently, we have established the large time behavior of the carbonation fronts mathematically. More precisely, we observe the existence of positive constants c and C such that c

≤ s(t) ≤ C

for t ≥ 0. This

t-law is usually used by engineers as the experimental law.

Goro Akagi (Kobe Universtiy, Japan)

Nonlinear diffusion equations involving p(x)-Laplacians

In this talk, we deal with nonlinear parabolic equations involving the so-called p(x)-Laplacians. We discuss well-posedness, large-time behaviors of solutions and limiting problems as p(x) tends to infinity in the whole or a part of domain. Our method of analysis is based on variational analysis and subdifferential calculus. This talk is based on a joint work with Kei Matsuura (Waseda University).

In this talk we use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heat flow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

In atmosphere-ocean science, the representation (or parameterization) of subgrid scale processes in numerical models is a notoriously difficult problem. Good representations (or lack thereof) of processes such as convection, cloud physics and surface interactions directly affect the quality and accuracy of forecasts and climate studies. In recent years, researchers have turned to stochastic methods in order to improve these parameterizations. I will discuss a stochastic, data-driven approach to the problem, in which unresolved processes are represented by collections of (data-inferred) Markov processes that depend on resoved model variables. This approach leads to hybrid stochastic-deterministic models consisting of differential equations coupled to Markov chains.

Odo Diekmann (Utrecht University)
(joint work with Mats Gyllenberg, Hans Metz and many others)

A delay equation is a rule for extending a function of time towards the future, on the basis of the known past. Renewal Equations prescribe the current value, while Delay Differential Equations prescribe the derivative of the current value. With a delay equation one can associate a dynamical system by translation along the extended function.

I will illustrate by way of examples how such equations arise in the description of the dynamics of structured populations and sketch the available theory, while making a plea for the development of numerical bifurcation tools.

Joep Evers (TU Eindhoven)
(joint work Adrian Muntean)

Modelling multi-component crowd dynamics via a two-scale, measure-theoretical approach

We present a framework to describe the dynamics of crowds, in which the behaviour of the crowd is considered from a two-fold perspective: both macroscopically (continuum) and microscopically (discrete). On both scales we specify mass measures and their transport, and we unify the micro and macro approaches in a single model.
We incorporate the ideas of mixture theory, which allows us to define several constituents (read: subpopulations) of the large crowd, each having its own partial velocity field. We thus have the possibility to examine the interactive behaviour between subgroups that have distinct characteristics. We especially aim at giving special properties to those pedestrians that are represented by the microscopic (discrete) part in the model. In real life situations they would play the role of leaders, terrorists, predators etc.
By identifying a suitable concept of entropy for the system, we derive an entropy inequality. From this inequality restrictions on the proposed velocity fields follow. Obeying these restrictions in the modelling phase, we make our assumptions more feasible. Under certain conditions, the entropy inequality even turns out to hold if we add anisotropy to the interactions. Anisotropy, in our model, means that the vision (or: perception) of individuals depends on the direction; people have a front and a back side.
Open questions concern the effect of leadership and anisotropy on the large-scale behaviour of the crowd. Do they influence the overall velocity dramatically, for instance?

Tasnim Fatima (TU Eindhoven)
(joint work with Adrian Muntean and Toyohiko Aiki)

Well-posedness of a nonlinear two-scale reaction-diffusion system modeling concrete corrosion
We consider a two-scale reaction-diffusion system modeling concrete corrosion in sewer pipes, which is defined on two different spatial scales. The system consists of four PDEs one of which is a porous medi- like equation, and an ODE. The system is nonlinear, partially dissipative, and coupled via the solid-water interface at the pore level and through the nonlinearity in the diffusion coefficient of the moisture. This model is obtained by upscaling of the physio-chemical processes taking place in the microstructure of the pipes. We show existence, uniqueness, L_^∞-bounds and positivity of the weak solutions. To prove the existence of weak solutions, we apply a two-scale Galerkin approach for a part of the system combined with a Schauder fixed-point argument for the moisture equation.

Well-posedness for the variational inequality related to the Navier-Stokes equations

In this talk, the existence problems for the Navier-Stokes equations with constraint are considered in the 3-dimensional space. These problems are motivated by an initial-boundary value problem for a thermohydraulics model in which the absolute value of the velocity field
is constrained. The abstract theory of nonlinear evolution equations governed by subdifferentials of time-dependent convex functionals is useful. In their mathematical treatment, the point of emphasis is to specify the class of time-dependence of convex functionals. First, well-posedness for the strong and weak variational inequalities are considered. Second, the existence problem for the Navier-Stokes equations with the temperature dependent constraint is considered. This is a joint work with professor Nobuyuki Kenmochi, Bukkyo University, Japan.

Junichi Harada (Waseda University, Japan)

Singularity of blow-up solutions to the heat equation with a nonlinear boundary condition

We study the blow-up profile of positive solutions the heat equation with a nonlinear boundary condition for the Sobolev subcritical case. In particular, we focus on spacial singularities of the blow-up profile.
To apply arguments in Herrero-Velazquez (92-93), we establish some estimates for a heat kernel of some linearized equation.

Charlotte K. Hemelrijk (University of Groningen)
(joint work with Hanno Hildenbrandt)

The variations in shape in the aerial displays of huge flocks of starlings (Sturnus vulgaris) above the sleeping site at dawn is amazing. Also other species of birds are highly variable in the shape of their flocks, more so than fish schools; fish schools are usually oblong. The causes of this variability, however, are hardly known. We investigate these causes with the help of a model of the self-organisation of a travelling group. In the present paper, we use a model, called StarDisplay, whose flocking patterns resemble qualitatively and quantitatively those of real birds, in particular starlings. In it, individuals coordinate with nearby others through attraction, alignment and avoidance (just as in models of fish schools). This is supplemented with some specifics of starling behaviour, namely 1) their aerial locomotion, 2) a low and constant number of interaction-partners and 3) preferential movement above a ‘sleeping area’.
As to shape, we measure the relative proportions of the flock and the longest dimension in respect of its direction of movement. We show that flock shape is usually more variable when local differences in movement in the flock are larger. This happens when a) flock size is larger, b) interacting partners are fewer, c) the flock turnings are stronger, d) individuals roll into the turn. In contrast to our expectations, when variability of speed in the flock is higher, its shape and the positions of its members are more static. We explain this and indicate the adaptive value of low variability of speed and spatial restriction of interaction and develop testable hypotheses.

Akio Ito (Kinki University, Japan)

Mathematical Model for Cardiomegaly of Spontaneously Hypertensive Rats

In this talk, we propose two mathematical models for cardiomegaly of a spontaneously hypertensive rat (SHR), which is one of the model animals to research the mechanism of various disease caused by hypertension. One model is a system of ODEs established from the viewpoint of system biology and we will give some numerical and experimental results. The other is a system of PDEs and ODEs, which takes into consideration a space distribution of some functional proteins. Finally, we give some idea to analyze our mathematical model by using the theory of quasivariational inequality.

Risei Kano (Kinki University, Japan)

The existence of solutions for tumor invasion models equipped with a diffusion depending on some stress

We consider the tumor invasion model of Chaplain – Anderson type with the effect of heat shock protein (HSP). This problem has the diffusion coefficient that is depend on some variables. In this talk, we discuss the result of tumor diffusion that is K_n=K_n(t,x,f) type.(time, place and unknown function ‘f’) This is an extension of the condition that K_n=K_n(t,x) has been studied. In addition, we introduce the results of computer simulations on models for the effect of HSP.

We learned a lot from serious disasters (earthquake, tsunami, typhoon,....) which had attacked some places in Japan. In most cases after disasters we face various problems, such as environment destroy for life and economic collapse, etc. It is quite important to recover human and economic activities in the disaster regions as soon as possible, by providing with suitable supports for them. In general it takes a long time (10 or 20 years) to recover the complete environment which is the basis of our life. Therefore, actually one has to restart our life in incomplete circumstances. Moreover, at the same time one has to start adequate reorganization of production systems in order to recover the economical situation in the disaster regions.
We know many mathematical models dealing with the evolution of human activity and the economic growth, independently each other. However, in disaster regions the both aspects should be treated simultaneously on a correlative connection, although the corresponding model is much more complicated than the above.
In this talk, we propose a simplified system of ordinary differential equations (ODEs), which we call the revival full model of human and economic activities in disaster regions. Our model consists of two systems (S) and (E) of ODEs. System (S) describes the evolution of environmental parameter s of our life, and (E) the economic growth parameter w in the disaster regions and non-disaster regions. One of crucial points in our modeling is how to express mathematically the correlative relationship between s and w as well as some sufficient condition for the self-reliant growth of w, namely the revival of economy. The full model is discussed in four periods: in the first period the main support is given by the public funds, in the second period it is given by both of the public and private (local) funds and in the third period it is given only by the private (local) funds. In the final period we expect to reach the economically stable situation, that is, the economic growth is achieved without any support, but the environment recovery is still in process of development.

Well- posedness of a mathematical model of carbon dioxide transport in concrete carbonation process

Concrete carbonation is a phenomenon that alkalinity by calcium hydroxide in cementitious material changes into acidity because of carbon dioxide in air. Concrete carbonation has been studied as a free boundary problem in one dimensional case so far. Recently, in three dimension case we propose a mathematical model of moisture transport in this process, and proved the existence of a time global solution of this model. As the second step for analyzing of the dynamics of concrete carbonation in three dimension, we focus on carbon dioxide transport. In this talk we propose a initial boundary value problem for a parabolic type equation as a mathematical model of carbon dioxide transport in this process, and give the existence results of a time global solution of this problem.

Cell-based modeling of angiogenic blood vessel sprouting: cell-ECM interaction and tip-cell selection

Angiogenesis, the growth of new blood vessels from existing ones, is a topic of intensive experimental investigation so its phenomenology, the participating cell types, and the molecular signals contributing to it have been well characterized. Yet it is poorly understood how these biological components fit together dynamically to drive the outgrowth of blood vessels. Cell-based simulation models of blood vessel growth describe the behavior of cells and the chemicals they produce. They then predict the collective behavior resulting from cell-cell interactions. Thus they help analyze how cells assemble into blood vessels, and reveal how cell behavior depends on the microenvironment the cells produce collectively.
Our simulation models, based on a Cellular Potts model combined with reaction-diffusion equations, have shown that the elongated shape of cells is key to correct spatiotemporal in silico replication of vascular network growth. We also identified a new stochastic mechanism for blood vessel sprouting. Here I will briefly discuss new insights into the role of cell shape and stochastic motility during vascular branching. Then I will present recent results on the role of tip cells, suggesting that tip cell-stalk cell interactions accelerate angiogenic sprouting. I will also discuss our recent cell-based modeling studies of cell-extracellular matrix interactions during angiogenesis.

Yusuke Murase (Meijo University, Japan)

Nonlinear evolution equations for mathematical modeling for brewing process of Japanese Sake

In this talk, we introduce our mathematical modeling for brewing process of Japanese Sake, and discuss solvability of nonlinear evolution equations which is corresponding to parabolic quasi-variational inequalities. Our evolution equations are generated by subdifferentials and some perturbations. In addition, we explain the relationship between the evolution equations and mathematical modeling.

In this talk, we consider a Cahn-Hilliard/Allen-Cahn equation, which was derived in Karali-Katsoulakis. This mean field partial differential equation contains qualitatively microscopic information on particle-particle interactions and multiple particle dynamics. In a suitable scaling, the singular limit of this equation is mean curvature flow, like Allen-Cahn equation, but with a different mobility. As a mathematical analysis, we improve the existence of solution for a standard double-well potential in dimension 1 to 4 in view of the free energy. We also mention about the stochastic version of this equation.

The diffusive VSC model describes the growth of fungal hyphae. It assumes that building materials for the cell wall are produced inside the cell and diffuse outwards, where they are assimilated causing the cell wall to expand orthogonally. This yields a geometric evolution equation for the surface of the cell. In this talk we will show how to prove the existence of travelling solutions, similar to those observed experimentally.

Takeshi Ohtsuka (Gunma University, Japan)

Stability of bunched steps in spiral crystal growth

Burton, Cabrera and Frank proposed the theory of the evolution of monomolecular spiral steps on the crystal surface in 1951. Kobayashi or Karma and Plapp proposed an Allen--Cahn type equation, and Smereka or the speaker proposed a level set formulation for the evolution of spiral curves, respectively. On the other hand, one can find the spiral patterns by multiple molecular height of steps on crystal surface in physical experiences. Thus, the stability of bunched steps is concerned.
However, instability of bunched steps by Allen--Cahn type equation are derived by Ogiwara and Nakamura.
In this talk we consider evolution of bunched spiral curves with the level set formulation by the speaker. Our model is derived from asymptotic expansion of the Allen--Cahn type equation.
However, we prove the stability of bunched steps in the level set equation.

Permanent deformation of metals results from the collective motion of a large number of dislocations. These dislocations are defects in the atomic lattice, and their motion causes the atomic layers to slide over each other. Many of these atomic-scale sliding events may combine to produce a macroscopic change of geometry that we observe as permanent deformation.
Although good models are available both at the level of the atomic lattice and at the level of continuum mechanics, there currently is no method, formal or rigorous, that allows us to take a limit and connect the different scales in a convincing way. This is a major obstacle for the design and engineering of high-performance steels.
In this talk I describe recent progress in this area, together with Lucia Scardia, Marc Geers, and Ron Peerlings. I will show how a simplified setup produces new insight into this difficult connection, and provides a hint as to how we should think about the more general case.

In this presentation we focus on a mathematical model for dissolution and precipitation in porous media. The model fits into a generic framework applicable in various contexts, including bio-remediation, drug release, battery modelling, weathering of stones or concrete carbonation. Compared to similar models for reactive flows, the particularity encountered here is in the modeling of the dissolution. This involves a multi-valued rate and can explain the occurrence of dissolution fronts.
After addressing some modeling details, we present results concerning the existence and uniqueness of a weak solution at the pore scale. Next, we employ upscaling techniques to derive the corresponding model at the core (laboratory) scale, for which
existence and uniqueness results are provided.

This is a joint work with C.J. van Duijn, K. Kumar (both Eindhoven), A. Mikelic (Lyon), T.L. van Noorden, M. Neuss-Radu (both Erlangen), and F.A. Radu (Bergen)

We discuss a 2D moving-boundary problem for the Laplacian with Robin boundary conditions in an exterior domain. It arises as model for Hele-Shaw ﬂow of a bubble with kinetic undercooling regularization and is also discussed in the context of models for electrical streamer discharges.

The corresponding evolution equation is given by a degenerate, nonlinear transport problem with nonlocal lower-order dependence. We identify the local structure of the set of traveling-wave solutions in the vicinity of trivial (circular) ones. We ﬁnd that there is a unique nontrivial traveling wave for each velocity near the trivial one. Therefore, the trivial solutions are unstable in a comoving frame.

The degeneracy of our problem is reﬂected in a loss of regularity in the estimates for the linearization. Moreover, there is an upper bound for the regularity of its solutions.

We consider spatio-temporal pattern formation in spatially one-dimensional Landau-Lifschitz-Gilbert equations modelling the evolution of magnetization in materials. We prove existence of a rich class of patterns composed of wavetrains and/or homogeneous states, and also discuss stability. This is joint work with Christof Melcher (RWTH Aachen).

Motivated by so-called Gibbs-non-Gibbs transitions, we study diffusion processes with small variance starting from an initial measure that satisfies the large deviation principle, and conditioned to end at a fixed point in the future. We show that for large times, the optimal trajectories can be non-unique. The same behavior is shown for birth and death processes close to their deterministic limit.

On the norm-inflation solutions to the Navier-Stokes equations in the critical space

The ill-posedness of the incompressible 3-D Navier-Stokes equations in the critical Besov space is concerned. A lack of the equicontinuity was shown by Bourgain-Pavlovic, using the norm-inflation argument of mild solutions (solutions of the integral equation via the heat semigroup). In this talk, the term-wise estimates for the sequence of successive approximation of a mild solution are derived. This leads us to a rigorous proof of the ill-posedness theory. Moreover, from the same ideas of Bourgain-Pavlovic, the necessity of taking a `good' subsequence of the successive approximation is also proved. Indeed, there exists a `bad' subsequece who diverges.

Ken Shirakawa (Chiba University, Japan)

Energy-dissipative solution to one-dimensional phase field model of grain boundary

This study is a joint work with Dr. Hiroshi Watanabe, Salesian Polytechnic, Japan.
In this talk, a one-dimensional system of two parabolic type equations is considered. This system is based on a phase field model of grain boundary, proposed by Kobayashi-Warren-Carter [Physica D, 140 (2000), 141-150], and it is derived as a gradient flow of a governing energy, called free-energy.
The focus in this talk is on a special kind of solution, named as "energy-dissipative solution", which realizes the dissipation of the free-energy in time. Consequently, the existence of energy-dissipative solution and some related topics will be presented as the main results of this talk.

Gierer-Meinhardt system describes morphogenesis of hydra in the context of Turing patterns. There is a parameter region where the ODE part takes periodic orbits. If two diffusion coefficients are comparable in this parameter region, then any solution exists global-in-time and is absorbed into one of the ODE orbit. An underlying variational structure is revealed applicable to other biological models; joint work with G. Karali and Y. Yamada.

Souhei Tasaki (Osaka University, Japan)
(joint work with Irena Pawlow and Takashi Suzuki)

We study a strain-gradient type thermoviscoelastic system. We focus on the stationary states and their dynamical stability. The adiabatic stationary state is formulated as a nonlinear eigenvalue problem with non-local terms associated with the total energy conservation. We reveal a unified structure, called semi-dualities, of the thermoviscoelastic system of viscosity-capillarity type with temperature-dependent viscous and elastic moduli. Based on the semi-dual structure we construct a series of general results concerning the stationary states and their stability. The application of these results together with the bifurcation theory allows to analyze the total set of the stationary solutions in more detail.

Tohru Wakasa (Meiji University, Japan)

Mathematical analysis for a simplified tumor growth model with contact-inhibition

In the last two decades several mathematical models on tumor growth has been discussed by many researchers. For a competying process between the normal cells and the abnormal cells, Chaplain, Graziano and Presiozi have proposed a macroscopic 5 component PDE model of reaction-diffusion type. From a viewpoint of contact-inhibition of cells, which is commonly observed for colony formation in vitro (and vivo), a simplified system for two cell populations has been introduced and analyzed by Bertsch, Dal-Passo and Mimura. This contact-inhibition model, shows us a segregated property: if the initial distribution for normal and abnormal cells are segregated from each other, the solution keeps segregated after the contact.
We are interested in a qualitative behavior of the solutions to the contact-inhibition model including segregating/overlapping properties. In this talk we consider the one-dimensional model, and based on numerical evidence for the initial value problem, mathematical results on a free boundary formulation and traveling wave solutions are given.
This talk is based on the joint work with Professors Michiel Bertsch and Masayasu Mimura.

Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients

Strongly degenerate parabolic equations are regarded as a linear combination of the time-dependent conservation laws (quasilinear hyperbolic equations) and the porous medium type equations (nonlinear degenerate parabolic equations). Thus, this equations has both properties of hyperbolic equation and those of parabolic equations and describes various nonlinear convective diffusion phenomena such as filtlation problems, Stefan problems and so on.
In this talk we consider that strongly degenerate parabolic equation has discontinuous coefficients. In particular, we forcus our attention on boundary value problems for this type of equations. In fact, we show the unique existence of weak solutions for the Neumann problem and the zero-flux boundary value problem. We explain construction of the weak solutions using compensated compactness method and H-measure.

A basic model for cell swelling by osmosis is constructed, resulting in a free boundary problem. For radially symmetric initial conditions, this model can be formulated as a gradient flow on a metric by choosing a suitable pair of functional and metric. This particular choice does not require the osmotic force to be included in the formulation explicitly. It appears that this result can be generalized to non-symmetric initial conditions.

Conference Location
The workshop location is EURANDOM, Den Dolech 2, 5612 AZ Eindhoven, Laplace Building, 1st floor, LG 1.105.

Contact
For more information please contact Mrs. Patty Koorn,
Workshop officer of EURANDOM

European Institute for
Statistics, Probability, Stochastic Operations Research
and their Applications

Last updated 28-11-11,
by PK

European Institute for Statistics, Probability, Stochastic Operations Research and their Applications

Last updated 28-11-11, by PK

European Institute for
Statistics, Probability, Stochastic Operations Research
and their Applications

Last updated 28-11-11,
by PK