The paper studies quasilinear elliptic problems in the Sobolev spaces W-1,W-p(Omega), Omega subset of R-N, with p = N, that is, the case of Pohozhaev-Trudinger-Moser inequality. Similarly to the case p < N where the loss of compactness in W-1,W-p(R-N) occurs due to dilation operators u bar right arrow t((N-p)/p)u(tx), t > 0, and can be accounted for in decompositions of the type of Struwe's "global compactness" and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W-0(1,N) over a ball in R-N. We give a one-parameter scale of Hardy-Sobolev functionals, a "p = N"-counterpart of the Holder interpolation scale, for p > N, between the Hardy functional integral vertical bar u vertical bar(p)/vertical bar x vertical bar(p) dx and the Sobolev functional integral vertical bar u vertical bar(pN/(N-mp)) dx. Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W-1,W-N-norms under Hardy-Sobolev constraints.

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r(-b) for functions in R-n. The exact Hardy constant c(b) = c(b) (n) is found and generalized minimizers are given. The constant cb vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in R-2. Analogous inequalities are proved in the case c(b) = 0 under constraints and, with error terms, for a bounded domain.

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the Trudinger-Moser inequality on the open unit disk B subset of R-2, recently proved by Mancini and Sandeep [g], (Arxiv 0910.0971). Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to the Mobius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity integral |u|(2)* in the higher dimension than the original Trudinger-Moser nonlinearity.

We study random graphs, both G(n, p) and G(n, m), with random orientations on the edges. For three fixed distinct vertices s, a, b we study the correlation, in the combined probability space, of the events {a -> s} and {s -> b}. For G(n, p), we prove that there is a p(c) = 1/2 such that for a fixed p < p(c) the correlation is negative for large enough n and for p > p(c) the correlation is positive for large enough n. We conjecture that for a fixed n >= 27 the correlation changes sign three times for three critical values of p. For G(n, m) it is similarly proved that, with p = m/((n)(2)), there is a critical p(c) that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any l directed edges in G(n, m), is thought to be of independent interest. We present exact recursions to compute P(a -> s) and P(a -> s, s -> b). We also briefly discuss the corresponding question in the quenched version of the problem.

This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations.

Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r₀)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough. 

Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough.

In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling.

Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case.

In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ.

Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype\begin{equation*}  \partial_tu= -\sum_{i=1}^{m}X_i^\ast ( |\X u|^{p-2} X_i u)\end{equation*}where $p\ge 2$, $ \ \X = (X_1,\ldots, X_m)$ is a system of Lipschitz vector fields defined on a smooth manifold $\M$ endowed with a Borel measure $\mu$, and $X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are derived assuming that (i) the control distance $d$ generated by $\X$ induces the same topology on $\M$; (ii) a doubling condition for the $\mu$-measure of $d-$metric balls and (iii) the validity of a Poincar\'e inequality involving $\X$ and $\mu$. Our results extend the recent work in \cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting including the model cases of (1) metrics generated by H\"ormander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

We develop algorithms for computations of Green's function and its Fourier coefficients, F-n(z, s), on Fuchsian groups with one cusp An analogue of a Rankin-Selberg bound for F-n(z,s) is presented.

We present some examples of numerical investigations of the value distribution of Green's function and of its Fourier coefficients on the modular group PSL(2,Z) Our results indicate that both Green's function G(s)(z, w) and its Fourier coefficients F-n(z,s) have a Gaussian value distribution in the semiclassical limit when Re s = 1/2.

The purpose of this paper is to investigate the connection between network evolution and technology embedding. To this end, we performed an exploratory case study of the network surrounding an eco-sustainable technology, Leaf House, Italy's first zero-carbon emission house. We apply theories on technological development within industrial networks, with a specific focus on their resource layer and on the three settings involved in embedding an innovation: “developing”, “producing”, and “using”. Our results contribute to these theories by developing four propositions on the connections between network evolution and embedding: first, technology embedding entails both downstream network expansion and upstream restrictions. Secondly, conflicts among actors increase as technology embedding approaches the producing and using settings. Third and fourth, the more the shapes a technology can assume, and the more each of these shapes involves actors acting in different settings, the easier it is to embed it. The paper concludes with managerial implications and suggestions for further research.

The paper establishes a functional version of the Hoeffding   combinatorial central limit theorem. First, a pre-limiting Gaussian   process approximation is defined, and is shown to be at a distance of   the order of the Lyapounov ratio from the original random process.   Distance is measured by comparison of expectations of smooth   functionals of the processes, and the argument is by way of Stein's   method. The pre-limiting process is then shown, under weak conditions,   to converge to a Gaussian limit process. The theorem is used to   describe the shape of random permutation tableaux.

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open.

We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments.

In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures.

We introduce the QDs in a sub domain of and Rⁿ study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper.

In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C¹-graph up to that part.

In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.

We study minimizers of the functional where B_{1}^{{\mathchoice {\raise .17ex\hbox {\scriptstyle +}} {\raise .17ex\hbox {\scriptstyle +}} {\raise .1ex\hbox {\scriptscriptstyle +}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\} , u=0 on {x∈B ₁:x ₁=0}, \lambda^{{\mathchoice {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .1ex\hbox {\scriptscriptstyle \pm }} {\scriptscriptstyle \pm }}} are two positive constants and 0<p<1. In two dimensions, we prove that the free boundary is a uniform C ¹ graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {x:±u(x)>0}.

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs, vertex random graphs are generalizations of geometric random graphs, and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.

Investigating how individuals are affected by environmental pollution is relatively straightforward, for example through conducting field studies or laboratory toxicity tests. Exploring such effects at a population level is considerably more difficult. Nonetheless, the exploration of population-level effects is important as the outcomes may differ from those seen at the individual level. Eelpout (Zoarces viviparus L) have been used for several years as a bioindicator for hazard substances in both the field and laboratory tests, and individual effects on reproduction have been reported. However, the influence of these effects at the population level remained unexplored. In this study, four Leslie matrix models were parameterized using data from non-polluted eelpout populations (Skagerrak, Baltic Proper, Gulf of Bothnia and Gulf of Finland). The four sites represent an environmental gradient in salinity. Furthermore, life-history data revealed differences between the sites with growth rate, fecundity, age at maturity and longevity being the most significant. The effect of pollution on natural eelpout populations was then simulated by combining the outputs from the Leslie matrices with data from laboratory and field studies exploring reproductive impairment in contaminated environments. Our results show that despite differences in life-history characteristics between sites, survival of early life stages (i.e. larvae and zero-year-old fish) was the most important factor affecting population growth and persistence for all sites. The range of change in survival of larvae necessary to change population dynamics (i.e. growth) and persistence is well within the range documented in recipient and experimental studies of chemicals and industrial waste waters. Overall, larval malformation resulting from environmental pollution can have large effects on natural populations, leading to population losses and possibly even extinction. This study hereby contributes valuable knowledge by extending individual-level effects of environmental contaminants to the population level. (C) 2012 Elsevier Inc. All rights reserved.

In 2007, we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this article is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with (conditionally) independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3-122 (2007), as well as related results of Bollobas, Borgs, Chayes and Riordan (Ann. Probab. 38:150-183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).

Given omega >= 1, let Z((omega))(2) be the graph with vertex Set Z(2) in which two vertices are joined if they agree in one coordinate and differ by at most omega in the other. (Thus Z((1))(2) is precisely Z(2).) Let p(c)(omega) be the critical probability for site percolation on Z((omega))(2) Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that lim(omega ->infinity) omega p(c)(omega) = log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

In this paper we study the minimal number tau(S, G) of translates of an arbitrary subset S of a group G needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, the worst-case efficiency when S has k elements is of order 1/log k. We show that if n(k) grows at a suitable rate with k, then almost every k-subset of any given group with order n comes close to this worst-case bound. In contrast, if n(k) grows very rapidly, or if k is fixed and n ->infinity, then almost every k-subset of the cyclic group with order n comes close to the optimal efficiency.

Proteins participating in a protein-protein interaction network can be grouped into homology classes following their common ancestry. Proteins added to the network correspond to genes added to the classes, so the dynamics of the two objects are intrinsically linked. Here we first introduce a statistical model describing the joint growth of the network and the partitioning of nodes into classes, which is studied through a combined mean-field and simulation approach. We then employ this unified framework to address the specific issue of the age dependence of protein interactions through the definition of three different node wiring or divergence schemes. A comparison with empirical data indicates that an age-dependent divergence move is necessary in order to reproduce the basic topological observables together with the age correlation between interacting nodes visible in empirical data. We also discuss the possibility of nontrivial joint partition and topology observables.

Members of social groups need to coordinate their behaviour when choosing between alternative activities. Consensus decisions enable group members to maintain group cohesion and one way to reach consensus is to rely on quorums. A quorum response is where the probability of an activity change sharply increases with the number of individuals supporting the new activity. Here, we investigated how meerkats (Suricata suricatta) use vocalizations in the context of movement decisions. Moving calls emitted by meerkats increased the speed of the group, with a sharp increase in the probability of changing foraging patch when the number of group members joining the chorus increased from two up to three. These calls had no apparent effect on the group's movement direction. When dominant individuals were involved in the chorus, the group's reaction was not stronger than when only subordinates called. Groups only increased speed in response to playbacks of moving calls from one individual when other group members emitted moving calls as well. The voting mechanism linked to a quorum probably allows meerkat groups to change foraging patches cohesively with increased speed. Such vocal coordination may reflect an aggregation rule linking individual assessment of foraging patch quality to group travel route.

We perform a numerical study of the breakdown of hyperbolicity of quasi-periodic attractors in the dissipative standard map. In this study, we compute the quasi-periodic attractors together with their stable and tangent bundles. We observe that the loss of normal hyperbolicity comes from the collision of the stable and tangent bundles of the quasi-periodic attractor. We provide numerical evidence that, close to the breakdown, the angle between the invariant bundles has a linear behavior with respect to the perturbing parameter. This linear behavior agrees with the universal asymptotics of the general framework of breakdown of hyperbolic quasi-periodic tori in skew product systems.

The Mahonian statistic is the number of inversions in a permutation of a multiset with a(i) elements of type i, 1 <= i <= m. The counting function for this statistic is the q analog of the multinomial coefficient (a(1) +...+4 a(m) a(1)...a(m)), and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is computer-assisted, based on the method of moments. The Maple package Mahoni anStat, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coefficients of the q-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside).

We obtain restriction results of K. De Leeuw's type for maximal operators defined through Fourier multipliers of either strong or weak type for weighted L-p spaces with 1 <= p <= infinity. Applications to the case of Hormander-Mihlin multipliers, singular integral operators and Bochner-Riesz sums are given.

We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form \fancyscriptL=∑i,j=1mai,j(z)∂xixj+∑i=1mai(z)∂xi+∑i,j=1Nbi,jxi∂xj−∂t, where z = (x, t) belongs to an open set Ω⊂RN×R , and 1 ≤ m ≤ N. Let z˜∈Ω , let K be a compact subset of Ω−− , and let Σ⊂∂Ω be such that K∩∂Ω⊂Σ . We give sufficient geometric conditions for the validity of the following Carleson type estimate. There exists a positive constant C _K , depending only on Ω,Σ,K,z˜ and on \fancyscriptL , such that supKu≤CKu(z˜), for every non-negative solution u of \fancyscriptLu=0 in Ω such that u∣Σ=0 .

We prove a Carleson type estimate, in Lipschitz type domains, for non-negative solutions to a class of second order degenerate differential operators of Kolmogorov type of the form L = Sigma(m)(i,j=1)a(i,j)(z)partial derivative x(i)x(j) + Sigma(m)(i=1)a(i)(z)partial derivative(xi) + Sigma(N)(i,j=1) b(i,j)x(i)partial derivative(xj) - partial derivative(t), where z = (x, t) is an element of RN+1, 1 <= m <= N. Our estimate is scale-invariant and generalizes previous results valid for second order uniformly parabolic equations to the class of operators considered.

Microbes produce many molecules that are important for their growth and development, and the exploitation of these secretions by nonproducers has recently become an important paradigm in microbial social evolution. Although the production of these public-goods molecules has been studied intensely, little is known of how the benefits accrued and the costs incurred depend on the quantity of public-goods molecules produced. We focus here on the relationship between the shape of the benefit curve and cellular density, using a model assuming three types of benefit functions: diminishing, accelerating, and sigmoidal (accelerating and then diminishing). We classify the latter two as being synergistic and argue that sigmoidal curves are common in microbial systems. Synergistic benefit curves interact with group sizes to give very different expected evolutionary dynamics. In particular, we show that whether and to what extent microbes evolve to produce public goods depends strongly on group size. We show that synergy can create an "evolutionary trap" that can stymie the establishment and maintenance of cooperation. By allowing density-dependent regulation of production (quorum sensing), we show how this trap may be avoided. We discuss the implications of our results on experimental design.

We study semilinear elliptic equations in a generally unbounded domain Omega subset of R-N when the pertinent quadratic form is nonnegative and the potential is generally singular, typically a homogeneous function of degree -2. We prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations, while the nonlinearity is a perturbation of a self-similar, possibly oscillating, term f(infinity) of critical growth satisfying f(infinity)(lambda(j)s)= lambda N+2/N-2 f(infinity)(s), j is an element of Z, s is an element of R. This paper focuses on two qualitatively different cases of this problem, one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space. In the latter case we allow nonlinearities with oscillatory critical growth. An important example of such quadratic form is the one on RN with the radial Hardy potential -mu vertical bar x vertical bar(-2) with mu = mu(*) in the first case, mu < mu(*) in the second case, where mu(*) = (N-2)(2)/4 is the largest constant for which the energy form remains nonnegative.

Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in C-n. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson (Bousquet-Melou and Janson, Ann Appl Probab 16 (2006) 1597-1632), saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion).

In a uniform random recursive k-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S (n) is the shortest path distance from node n to the root, then we determine the constant sigma such that S (n) /log n ->sigma in probability as n -> a. We also show that max (1a parts per thousand currency signia parts per thousand currency signn) S (i) /log n ->sigma in probability.

Recruitment via pheromone trails by ants is arguably one of the best-studied examples of self-organization in animal societies. Yet it is still unclear if and how trail recruitment allows a colony to adapt to changes in its foraging environment. We study foraging decisions by colonies of the ant Pheidole megacephala under dynamic conditions. Our experiments show that P. megacephala, unlike many other mass recruiting species, can make a collective decision for the better of two food sources even when the environment changes dynamically. We developed a stochastic differential equation model that explains our data qualitatively and quantitatively. Analysing this model reveals that both deterministic and stochastic effects (noise) work together to allow colonies to efficiently track changes in the environment. Our study thus suggests that a certain level of noise is not a disturbance in self-organized decision-making but rather serves an important functional role.

In this paper we investigate the foraging activity of an invasive ant species, the big headed ant Pheidole megacephala. We establish that the ants' behavior is consistent with the use of two different pheromone signals, both of which recruit nestmates. Our experiments suggest that during exploration the ants deposit a long-lasting pheromone that elicits a weak recruitment of nestmates, while when exploiting food the ants deposit a shorter lasting pheromone eliciting a much stronger recruitment. We further investigate experimentally the role of these pheromones under both static and dynamic conditions and develop a mathematical model based on the hypothesis that exploration locally enhances exploitation, while exploitation locally suppresses exploration. The model and the experiments indicate that exploratory pheromone allows the colony to more quickly mobilize foragers when food is discovered. Furthermore, the combination of two pheromones allows colonies to track changing foraging conditions more effectively than would a single pheromone. In addition to the already known causes for the ecological success of invasive ant species, our study suggests that their opportunistic strategy of rapid food discovery and ability to react to changes in the environment may have strongly contributed to their dominance over native species.

One of the key challenges in the study of networks is linking structure to function. For example, how do design requirements about the speed and accuracy with which information is transferred through a network determine its form?  We show that different strains of the slime mould Physarum polycephalum form different network structures, ranging from a diffuse network of thin links to a tree-like branching structure.  Using a current-reinforced random walk model, we explain these different structures in terms of two model parameters: the strength and the degree of non-linearity in the reinforcement. These parameters are further shown to tune the speed and accuracy with which the network can detect resource gradients. We use a battery of experimental tests to show that Physarum strains with diffuse networks make more accurate but slower decisions and those with thick, trunk branches make faster less accurate decisions. Intermediate structures can also be found which are relatively fast and accurate. The current reinforced random walk employed by the slime mould provides a tunable algorithm for decision-making, which may also apply in other systems where transport networks are constructed.

It is well known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the present paper we consider the inverse problem, that is, given prices of perpetual American options for different strikes, we show how to construct a time-homogeneous stock price model which reproduces the given option prices.

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.