@inbook{4417,
  abstract     = {Counterexample-guided abstraction refinement (CEGAR) is a powerful technique to scale automatic program analysis techniques to large programs. However, so far it has been used primarily for model checking in the context of predicate abstraction. We formalize CEGAR for general powerset domains. If a spurious abstract counterexample needs to be removed through abstraction refinement, there are often several choices, such as which program location(s) to refine, which abstract domain(s) to use at different locations, and which abstract values to compute. We define several plausible preference orderings on abstraction refinements, such as refining as “late” as possible and as “coarse” as possible. We present generic algorithms for finding refinements that are optimal with respect to the different preference orderings. We also compare the different orderings with respect to desirable properties, including the property if locally optimal refinements compose to a global optimum. Finally, we point out some difficulties with CEGAR for non-powerset domains.},
  author       = {Manevich, Roman and Field, John and Thomas Henzinger and Ramalingam, Ganesan and Sagiv, Mooly},
  booktitle    = {Program Analysis and Compilation, Theory and Practice: Essays Dedicated to Reinhard Wilhelm on the Occasion of His 60th Birthday},
  pages        = {273 -- 292},
  publisher    = {Springer},
  title        = {{Abstract counterexample-based refinement for powerset domains}},
  doi          = {10.1007/978-3-540-71322-7_13},
  volume       = {4444},
  year         = {2007},
}

@article{4446,
  abstract     = {The Embedded Machine is a virtual machine that mediates in real time the interaction between software processes and physical processes. It separates the compilation of embedded programs into two phases. The first phase, the platform-independent compiler phase, generates E code (code executed by the Embedded Machine), which supervises the timing, not the scheduling of, application tasks relative to external events such as clock ticks and sensor interrupts. E code is portable and, given an input behavior, exhibits predictable (i.e., deterministic) timing and output behavior. The second phase, the platform-dependent compiler phase, checks the time safety of the E code, that is, whether platform performance (determined by the hardware) and platform utilization (determined by the scheduler of the operating system) enable its timely execution. We have used the Embedded Machine to compile and execute high-performance control applications written in Giotto, such as the flight control system of an autonomous model helicopter.},
  author       = {Thomas Henzinger and Kirsch, Christoph M},
  journal      = {ACM Transactions on Programming Languages and Systems (TOPLAS)},
  number       = {393},
  publisher    = {ACM},
  title        = {{The embedded machine: Predictable, portable real-time code}},
  doi          = {10.1145/1286821.1286824},
  volume       = {29},
  year         = {2007},
}

@inproceedings{4511,
  abstract     = {In the traditional view, a language is a set of words, i.e., a function from words to boolean values. We call this view “qualitative,” because each word either belongs to or does not belong to a language. Let Σ be an alphabet, and let us consider infinite words over Σ. Formally, a qualitative language over Σ is a function A: B . There are many applications of qualitative languages. For example, qualitative languages are used to specify the legal behaviors of systems, and zero-sum objectives of games played on graphs. In the former case, each behavior of a system is either legal or illegal; in the latter case, each outcome of a game is either winning or losing. For defining languages, it is convenient to use finite acceptors (or generators). In particular, qualitative languages are often defined using finite-state machines (so-called ω-automata) whose transitions are labeled by letters from Σ. For example, the states of an ω-automaton may represent states of a system, and the transition labels may represent atomic observables of a behavior. There is a rich and well-studied theory of finite-state acceptors of qualitative languages, namely, the theory of theω-regular languages.},
  author       = {Thomas Henzinger},
  pages        = {20 -- 22},
  publisher    = {Springer},
  title        = {{Quantitative generalizations of languages}},
  doi          = {10.1007/978-3-540-73208-2_2},
  volume       = {4588},
  year         = {2007},
}

@inproceedings{4514,
  abstract     = {Digital technology is increasingly deployed in safety-critical situations. This calls for systematic design and verification methodologies that can cope with three major sources of system complexity: concurrency, real time, and uncertainty. We advocate a two-step process: formal modeling followed by algorithmic analysis (or, “model building” followed by “model checking”). We model the concurrent components of a reactive system as potential collaborators or adversaries in a multi-player game with temporal objectives, such as system safety. The real-time aspect of embedded systems requires models that combine discrete state transitions and continuous state evolutions. Uncertainty in the environment is naturally modeled by probabilistic state changes. As a result, we obtain three orthogonal extensions of the basic state-transition graph model for reactive systems —game graphs, timed graphs, and stochastic graphs— as well as combinations thereof. In this short text, we provide a uniform exposition of the underlying definitions. For verification algorithms, we refer the reader to the literature.},
  author       = {Thomas Henzinger},
  pages        = {103 -- 110},
  publisher    = {Springer},
  title        = {{Games, time, and probability: Graph models for system design and analysis}},
  doi          = {10.1007/978-3-540-69507-3_7},
  volume       = {4362},
  year         = {2007},
}

@article{4529,
  abstract     = {Computational modeling of biological systems is becoming increasingly important in efforts to better understand complex biological behaviors. In this review, we distinguish between two types of biological models—mathematical and computational—which differ in their representations of biological phenomena. We call the approach of constructing computational models of biological systems 'executable biology', as it focuses on the design of executable computer algorithms that mimic biological phenomena. We survey the main modeling efforts in this direction, emphasize the applicability and benefits of executable models in biological research and highlight some of the challenges that executable biology poses for biology and computer science. We claim that for executable biology to reach its full potential as a mainstream biological technique, formal and algorithmic approaches must be integrated into biological research. This will drive biology toward a more precise engineering discipline.},
  author       = {Fisher, Jasmin and Thomas Henzinger},
  journal      = {Nature Biotechnology},
  pages        = {1239 -- 1249},
  publisher    = {Nature Publishing Group},
  title        = {{Executable cell biology}},
  doi          = {10.1038/nbt1356},
  volume       = {25},
  year         = {2007},
}

@proceedings{4530,
  abstract     = {This book constitutes the refereed proceedings of the 21st International Workshop on Computer Science Logic, CSL 2007, held as the 16th Annual Conference of the EACSL in Lausanne, Switzerland. The 36 revised full papers presented together with the abstracts of six invited lectures are organized in topical sections on logic and games, expressiveness, games and trees, logic and deduction, lambda calculus, finite model theory, linear logic, proof theory, and game semantics.},
  author       = {Duparc, Jacques and Thomas Henzinger},
  booktitle    = {CSL: Computer Science Logic},
  publisher    = {Springer},
  title        = {{CSL: Computer Science Logic }},
  doi          = {10.1007/978-3-540-74915-8},
  volume       = {4646},
  year         = {2007},
}

@article{4531,
  abstract     = {Caenorhabditis elegans vulval development provides an important paradigm for studying the process of cell fate determination and pattern formation during animal development. Although many genes controlling vulval cell fate specification have been identified, how they orchestrate themselves to generate a robust and invariant pattern of cell fates is not yet completely understood. Here, we have developed a dynamic computational model incorporating the current mechanistic understanding of gene interactions during this patterning process. A key feature of our model is the inclusion of multiple modes of crosstalk between the epidermal growth factor receptor (EGFR) and LIN-12/Notch signaling pathways, which together determine the fates of the six vulval precursor cells (VPCs). Computational analysis, using the model-checking technique, provides new biological insights into the regulatory network governing VPC fate specification and predicts novel negative feedback loops. In addition, our analysis shows that most mutations affecting vulval development lead to stable fate patterns in spite of variations in synchronicity between VPCs. Computational searches for the basis of this robustness show that a sequential activation of the EGFR-mediated inductive signaling and LIN-12 / Notch-mediated lateral signaling pathways is key to achieve a stable cell fate pattern. We demonstrate experimentally a time-delay between the activation of the inductive and lateral signaling pathways in wild-type animals and the loss of sequential signaling in mutants showing unstable fate patterns; thus, validating two key predictions provided by our modeling work. The insights gained by our modeling study further substantiate the usefulness of executing and analyzing mechanistic models to investigate complex biological behaviors.},
  author       = {Fisher, Jasmin and Piterman, Nir and Hajnal, Alex and Thomas Henzinger},
  journal      = {PLoS Computational Biology},
  publisher    = {Public Library of Science},
  title        = {{Predictive modeling of signaling crosstalk during C. elegans vulval development}},
  doi          = {10.1371/journal.pcbi.0030092},
  volume       = {3(5):e92},
  year         = {2007},
}

@inproceedings{4537,
  abstract     = {The classical synthesis problem for reactive systems asks, given a proponent process A and an opponent process B, to refine A so that the closed-loop system A parallel to B satisfies a given specification Phi. The solution of this problem requires the computation of a winning strategy for proponent A in a game against opponent B. We define and study the co-synthesis problem, where the proponent A consists itself of two independent processes, A = A(1)parallel to A(2), with specifications Phi(1) and Phi(2), and the goal is to refine both A(1) and A(2) so that A(1)parallel to A(2)parallel to B satisfies Phi(1) boolean AND Phi(2). For example, if the opponent B is a fair scheduler for the two processes A(1) and A(2), and Phi(i) specifies the requirements of mutual exclusion for A(i) (e.g., starvation freedom), then the co-synthesis problem asks for the automatic synthesis of a mutual-exclusion protocol. We show that co-synthesis defined classically, with the processes A(1) and A(2) either collaborating or competing, does not capture desirable solutions. Instead, the proper formulation of co-synthesis is the one where process A, competes with A(2) but not at the price of violating Phi(1), and vice versa. We call this assume-guarantee synthesis and show that it can be solved by computing secure-equilibrium strategies. In particular, from mutual-exclusion requirements the assume-guarantee synthesis algorithm automatically computes Peterson's protocol.},
  author       = {Krishnendu Chatterjee and Thomas Henzinger},
  pages        = {261 -- 275},
  publisher    = {Springer},
  title        = {{Assume-guarantee synthesis}},
  doi          = {10.1007/978-3-540-71209-1_21},
  volume       = {4424},
  year         = {2007},
}

@article{4547,
  abstract     = {We study observation-based strategies for two-player turn-based games on graphs with omega-regular objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for any omega-regular objective. The fixed point is computed in the lattice of antichains of state sets. This algorithm has the advantages of being directed by the objective and of avoiding an explicit subset construction on the game graph. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomized observation-based strategy for a Buechi objective. This set is of interest because in the absence of perfect information, randomized strategies are more powerful than deterministic ones. We show that our algorithms are optimal by proving matching lower bounds.},
  author       = {Krishnendu Chatterjee and Doyen, Laurent and Thomas Henzinger and Raskin, Jean-François},
  journal      = {Logical Methods in Computer Science},
  number       = {184},
  pages        = {1 -- 23},
  publisher    = {International Federation of Computational Logic},
  title        = {{Algorithms for omega-regular games with imperfect information}},
  doi          = {10.2168/LMCS-3(3:4)2007},
  volume       = {3},
  year         = {2007},
}

@article{2657,
  abstract     = {The highest densities of the two metabotropic GABA subunits, GABA B1 and GABAB2, have been reported as occurring around the glutamatergic synapses between Purkinje cell spines and parallel fibre varicosities. In order to determine how this distribution is achieved during development, we investigated the expression pattern and the cellular and subcellular localization of the GABAB1 and GABAB2 subunits in the rat cerebellum during postnatal development. At the light microscopic level, immunoreactivity for the GABAB1 and GABAB2 subunits was very prominent in the developing molecular layer, especially in Purkinje cells. Using double immunofluorescence, we demonstrated that GABAB1 was transiently expressed in glial cells. At the electron microscopic level, immunoreactivity for GABAB receptors was always detected both pre- and postsynaptically. Presynaptically, GABAB1 and GABAB2 were localized in the extrasynaptic membrane of parallel fibres at all ages, and only rarely in GABAergic axons. Postsynaptically, GABAB receptors were localized to the extrasynaptic and perisynaptic plasma membrane of Purkinje cell dendrites and spines throughout development. Quantitative analysis and three-dimensional reconstructions further revealed a progressive developmental movement of the GABAB1 subunit on the surface of Purkinje cells from dendritic shafts to its final destination, the dendritic spines. Together, these results indicate that GABAB receptors undergo dynamic regulation during cerebellar development in association with the establishment and maturation of glutamatergic synapses to Purkinje cells.},
  author       = {Luján, Rafael and Ryuichi Shigemoto},
  journal      = {European Journal of Neuroscience},
  number       = {6},
  pages        = {1479 -- 1490},
  publisher    = {Wiley-Blackwell},
  title        = {{Localization of metabotropic GABA receptor subunits GABAB1 and GABAB2 relative to synaptic sites in the rat developing cerebellum}},
  doi          = {10.1111/j.1460-9568.2006.04669.x},
  volume       = {23},
  year         = {2006},
}

@article{2659,
  abstract     = {Transmembrane AMPA receptor regulatory proteins (TARPs), including stargazin/γ-2, are associated with AMPA receptors and participate in their surface delivery and anchoring at the postsynaptic membrane. TARPs may also act as a positive modulator of the AMPA receptor ion channel function; however, little is known about other TARP members except for stargazin/γ-2. We examined the synaptic localization of stargazin/γ-2 and γ-8 by immunoelectron microscopy and biochemical analysis. The analysis of sodium dodecyl sulfate-digested freeze-fracture replica labeling revealed that stargazin/γ-2 was concentrated in the postsynaptic area, whereas γ-8 was distributed both in synaptic and extra-synaptic plasma membranes of the hippocampal neuron. When a synaptic plasma membrane-enriched brain fraction was treated with Triton X-100 and separated by sucrose density gradient ultracentrifugation, a large proportion of NMDA receptor and stargazin/γ-2 was accumulated in raft-enriched fractions, whereas AMPA receptor and γ-8 were distributed in both the raft-enriched fractions and other Triton-insoluble fractions. Phosphorylation of stargazin/γ-2 and γ-8 was regulated by different sets of kinases and phosphatases in cultured cortical neurons. These results suggested that stargazin/γ-2 and γ-8 have distinct roles in postsynaptic membranes under the regulation of different intracellular signaling pathways.},
  author       = {Inamura, Mihoko and Itakura, Makoto and Okamoto, Hirotsugu and Hoka, Sumio and Mizoguchi, Akira and Fukazawa, Yugo and Ryuichi Shigemoto and Yamamori, Saori and Takahashi, Masami},
  journal      = {Neuroscience Research},
  number       = {1},
  pages        = {45 -- 53},
  publisher    = {Elsevier},
  title        = {{ Differential localization and regulation of stargazin-like protein, γ-8 and stargazin in the plasma membrane of hippocampal and cortical neurons}},
  doi          = {10.1016/j.neures.2006.01.004},
  volume       = {55},
  year         = {2006},
}

@article{2660,
  abstract     = {Pavlovian fear conditioning, a simple form of associative learning, is thought to involve the induction of associative, NMDA receptor-dependent long-term potentiation (LTP) in the lateral amygdala. Using a combined genetic and electrophysiological approach, we show here that lack of a specific GABAB receptor subtype, GABAB(1a,2), unmasks a nonassociative, NMDA receptor-independent form of presynaptic LTP at cortico-amygdala afferents. Moreover, the level of presynaptic GABA B(1a,2) receptor activation, and hence the balance between associative and nonassociative forms of LTP, can be dynamically modulated by local inhibitory activity. At the behavioral level, genetic loss of GABA B(1a) results in a generalization of conditioned fear to nonconditioned stimuli. Our findings indicate that presynaptic inhibition through GABAB(1a,2) receptors serves as an activity-dependent constraint on the induction of homosynaptic plasticity, which may be important to prevent the generalization of conditioned fear.},
  author       = {Shaban, Hamdy and Humeau, Yann and Herry, Cyril and Cassasus, Guillaume and Ryuichi Shigemoto and Ciocchi, Stéphane and Barbieri, Samuel and Van Der Putten, Herman V and Kaupmann, Klemens and Bettler, Bernhard and Lüthi, Andreas},
  journal      = {Nature Neuroscience},
  number       = {8},
  pages        = {1028 -- 1035},
  publisher    = {Nature Publishing Group},
  title        = {{Generalization of amygdala LTP and conditioned fear in the absence of presynaptic inhibition}},
  doi          = {10.1038/nn1732},
  volume       = {9},
  year         = {2006},
}

@article{2661,
  abstract     = {GABAB receptors are the G protein-coupled receptors for the main inhibitory neurotransmitter in the brain, γ-aminobutyric acid (GABA). Molecular diversity in the GABAB system arises from the GABAB1a and GABAB1b subunit isoforms that solely differ in their ectodomains by a pair of sushi repeats that is unique to GABAB1a. Using a combined genetic, physiological, and morphological approach, we now demonstrate that GABAB1 isoforms localize to distinct synaptic sites and convey separate functions in vivo. At hippocampal CA3-to-CA1 synapses, GABAB1a assembles heteroreceptors inhibiting glutamate release, while predominantly GABAB1b mediates postsynaptic inhibition. Electron microscopy reveals a synaptic distribution of GABAB1 isoforms that agrees with the observed functional differences. Transfected CA3 neurons selectively express GABAB1a in distal axons, suggesting that the sushi repeats, a conserved protein interaction motif, specify heteroreceptor localization. The constitutive absence of GABAB1a but not GABAB1b results in impaired synaptic plasticity and hippocampus-dependent memory, emphasizing molecular differences in synaptic GABAB functions.},
  author       = {Vigot, Réjan and Barbieri, Samuel and Bräuner-Osborne, Hans and Tureček, Rostislav and Ryuichi Shigemoto and Zhang, Yan Ping and Luján, Rafael and Jacobson, Laura H and Biermann, Barbara and Fritschy, Jean-Marc and Vacher, Claire-Marie and Müller, Matthias P and Sansig, Gilles and Guetg, Nicole and Cryan, John F and Kaupmann, Klemens and Gassmann, Martin and Oertner, Thomas G and Bettler, Bernhard},
  journal      = {Neuron},
  number       = {4},
  pages        = {589 -- 601},
  publisher    = {Elsevier},
  title        = {{Differential Compartmentalization and Distinct Functions of GABAB Receptor Variants}},
  doi          = {10.1016/j.neuron.2006.04.014},
  volume       = {50},
  year         = {2006},
}

@article{2662,
  abstract     = {G-protein-coupled inwardly rectifying K+ channels (Kir3 channels) coupled to metabotropic GABAB receptors are essential for the control of neuronal excitation. To determine the distribution of Kir3 channels and their spatial relationship to GABAB receptors on hippocampal pyramidal cells, we used a high-resolution immunocytochemical approach. Immunoreactivity for the Kir3.2 subunit was most abundant postsynaptically and localized to the extrasynaptic plasma membrane of dendritic shafts and spines of principal cells. Quantitative analysis of immunogold particles for Kir3.2 revealed an enrichment of the protein around putative glutamatergic synapses on dendritic spines, similar to that of GABA B1. Consistent with this observation, a high degree of coclustering of Kir3.2 and GABAB1 was revealed around excitatory synapses by the highly sensitive SDS-digested freeze-fracture replica immunolabeling. In contrast, in dendritic shafts receptors and channels were found to be mainly segregated. These results suggest that Kir3.2-containing K+ channels on dendritic spines preferentially mediate the effect of GABA, whereas channels on dendritic shafts are likely to be activated by other neurotransmitters as well. Thus, Kir3 channels, localized to different subcellular compartments of hippocampal principal cells, appear to be differentially involved in synaptic integration in pyramidal cell dendrites.},
  author       = {Kulik, Ákos and Vida, Imre and Fukazawa, Yugo and Guetg, Nicole and Kasugai, Yu and Marker, Cheryl L and Rigato, Franck and Bettler, Bernhard and Wickman, Kevin D and Frotscher, Michael and Ryuichi Shigemoto},
  journal      = {Journal of Neuroscience},
  number       = {16},
  pages        = {4289 -- 4297},
  publisher    = {Society for Neuroscience},
  title        = {{Compartment-dependent colocalization of Kir3.2-containing K+ channels and GABAB receptors in hippocampal pyramidal cells}},
  doi          = {10.1523/JNEUROSCI.4178-05.2006},
  volume       = {26},
  year         = {2006},
}

@article{2663,
  abstract     = {The rocker mice are hereditary ataxic mutants that carry a point mutation in the gene encoding the CaV2.1 (P/Q-type) Ca2+ channel α1 subunit, and show the mildest symptoms among the reported CaV2.1 mutant mice. We studied the basic characteristics of the rocker mutant Ca2+ channel and their impacts on excitatory synaptic transmission in cerebellar Purkinje cells (PCs). In acutely dissociated PC somas, the rocker mutant channel showed a moderate reduction in Ca2+ channel current density, whereas its kinetics and voltage dependency of gating remained nearly normal. Despite the small changes in channel function, synaptic transmission in the parallel fiber (PF)-PC synapses was severely impaired. The climbing fiber inputs onto PCs showed a moderate impairment but could elicit normal complex spikes. Presynaptic function of the PF-PC synapses, however, was unexpectedly almost normal in terms of paired-pulse facilitation, sensitivity to extracellular Ca2+ concentration and glutamate concentration in synaptic clefts. Electron microscopic analyses including freeze-fracture replica labeling revealed that both the number and density of postsynaptic α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors substantially decreased without gross structural changes of the PF-PC synapses. We also observed an abnormal arborization of PC dendrites in young adult rocker mice (∼ 1 month old). These lines of evidence suggest that even a moderate dysfunction of CaV2.1 Ca2+ channel can cause substantial changes in postsynaptic molecular composition of the PF-PC synapses and dendritic structure of PCs.},
  author       = {Kodama, Takashi and Itsukaichi-Nishida, Yuko and Fukazawa, Yugo and Wakamori, Minoru and Miyata, Mariko and Molnár, Elek and Mori, Yasuo and Ryuichi Shigemoto and Imoto, Keiji},
  journal      = {European Journal of Neuroscience},
  number       = {11},
  pages        = {2993 -- 3007},
  publisher    = {Wiley-Blackwell},
  title        = {{A CaV2.1 calcium channel mutation rocker reduces the number of postsynaptic AMPA receptors in parallel fiber-Purkinje cell synapses}},
  doi          = {10.1111/j.1460-9568.2006.05191.x},
  volume       = {24},
  year         = {2006},
}

@misc{2664,
  abstract     = {Metabotropic glutamate receptors (mGlus) are a family of G-protein-coupled receptors activated by the neurotransmitter glutamate. Molecular cloning has revealed eight different subtypes (mGlu1-8) with distinct molecular and pharmacological properties. Multiplicity in this receptor family is further generated through alternative splicing. mGlus activate a multitude of signalling pathways important for modulating neuronal excitability, synaptic plasticity and feedback regulation of neurotransmitter release. In this review, we summarize anatomical findings (from our work and that of other laboratories) describing their distribution in the central nervous system. Recent evidence regarding the localization of these receptors in peripheral tissues will also be examined. The distinct regional, cellular and subcellular distribution of mGlus in the brain will be discussed in view of their relationship to neurotransmitter release sites and of possible functional implications.},
  author       = {Ferraguti, Francesco and Ryuichi Shigemoto},
  booktitle    = {Cell and Tissue Research},
  number       = {2},
  pages        = {483 -- 504},
  publisher    = {Springer},
  title        = {{Metabotropic glutamate receptors}},
  doi          = {10.1007/s00441-006-0266-5},
  volume       = {326},
  year         = {2006},
}

@article{2745,
  abstract     = {We consider the dynamics of N boson systems interacting through a pair potential N -1 V a (x i -x j ) where V a (x)=a -3 V(x/a). We denote the solution to the N-particle Schrödinger equation by Ψ N, t . Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if u t solves the GP equation, then the family of k-particle density matrices [InlineMediaObject not available: see fulltext.] solves the GP hierarchy. Under the assumption that a = Nε for 0 < ε < 3/5, we prove that as N→∞ the limit points of the k-particle density matrices of Ψ N, t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫ V (x)dx. The uniqueness of the solutions of this hierarchy remains an open question.},
  author       = {Elgart, Alexander and László Erdös and Schlein, Benjamin and Yau, Horng-Tzer},
  journal      = {Archive for Rational Mechanics and Analysis},
  number       = {2},
  pages        = {265 -- 283},
  publisher    = {Springer},
  title        = {{Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons}},
  doi          = {10.1007/s00205-005-0388-z},
  volume       = {179},
  year         = {2006},
}

@inproceedings{2746,
  abstract     = {We consider random Schrödinger equations on Rd or Zd for d ≥ 3 with uncorrelated, identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0.},
  author       = {László Erdös and Salmhofer, Manfred and Yau, Horng-Tzer},
  pages        = {233 -- 257},
  publisher    = {World Scientific Publishing},
  title        = {{Towards the quantum Brownian motion}},
  doi          = {10.1007/3-540-34273-7_18},
  volume       = {690},
  year         = {2006},
}

@article{2747,
  abstract     = {Consider a system of N bosons on the three-dimensional unit torus interacting via a pair potential N 2V(N(x i - x j)) where x = (x i, . . ., x N) denotes the positions of the particles. Suppose that the initial data ψ N,0 satisfies the condition 〈ψ N,0, H 2 Nψ N,0) ≤ C N 2 where H N is the Hamiltonian of the Bose system. This condition is satisfied if ψ N,0 = W Nφ N,t where W N is an approximate ground state to H N and φ N,0 is regular. Let ψ N,t denote the solution to the Schrödinger equation with Hamiltonian H N. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if u t solves the GP equation, then the family of k-particle density matrices ⊗ k |u t?〉 〈 t | solves the GP hierarchy. We prove that as N → ∞ the limit points of the k-particle density matrices of ψ N,t are solutions of the GP hierarchy. Our analysis requires that the N-boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n.},
  author       = {László Erdös and Schlein, Benjamin and Yau, Horng-Tzer},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {12},
  pages        = {1659 -- 1741},
  publisher    = {Wiley-Blackwell},
  title        = {{Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate}},
  doi          = {10.1002/cpa.20123},
  volume       = {59},
  year         = {2006},
}

@article{2791,
  abstract     = {Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states, or it can occur suddenly. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion. Extensive calculations and experimental studies have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control.},
  author       = {Björn Hof and Westerweel, Jerry and Schneider, Tobias M and Eckhardt, Bruno},
  journal      = {Nature},
  number       = {7107},
  pages        = {59 -- 62},
  publisher    = {Nature Publishing Group},
  title        = {{Finite lifetime of turbulence in shear flows}},
  doi          = {10.1038/nature05089},
  volume       = {443},
  year         = {2006},
}