Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments

We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a program proposed by Lions and Souganidis that previously yielded homogenization in probability. The class of problems we study is strongly motivated by Sznitman's study of the quenched large deviations of Brownian motion interacting with a Poissonian potential, but applies to a general class of problems which are not amenable to probabilistic tools.Comment: 51 pages, 2 figures. We have added material and made some corrections to our previous versio

Vital dye labelling demonstrates a sacral neural crest contribution to the enteric nervous system of chick and mouse embryos

We have used the vital dye, DiI, to analyze the contribution of sacral neural crest cells to the enteric nervous system in chick and mouse embryos. In order to label premigratory sacral neural crest cells selectively, DiI was injected into the lumen of the neural tube at the level of the hindlimb. In chick embryos, DiI injections made prior to stage 19 resulted in labelled cells in the gut, which had emerged from the neural tube adjacent to somites 29–37. In mouse embryos, neural crest cells emigrated from the sacral neural tube between E9 and E9.5. In both chick and mouse embryos, DiI-labelled cells were observed in the rostral half of the somitic sclerotome, around the dorsal aorta, in the mesentery surrounding the gut, as well as within the epithelium of the gut. Mouse embryos, however, contained consistently fewer labelled cells than chick embryos. DiI-labelled cells first were observed in the rostral and dorsal portion of the gut. Paralleling the maturation of the embryo, there was a rostral-to-caudal sequence in which neural crest cells populated the gut at the sacral level. In addition, neural crest cells appeared within the gut in a dorsal-to-ventral sequence, suggesting that the cells entered the gut dorsally and moved progressively ventrally. The present results resolve a long-standing discrepancy in the literature by demonstrating that sacral neural crest cells in both the chick and mouse contribute to the enteric nervous system in the postumbilical gut

A vital dye analysis of the timing and pathways of avian trunk neural crest cell migration

To permit a more detailed analysis of neural crest cell migratory pathways in the chick embryo, neural crest cells were labelled with a nondeleterious membrane intercalating vital dye, DiI. All neural tube cells with endfeet in contact with the lumen, including premigratory neural crest cells, were labelled by pressure injecting a solution of DiI into the lumen of the neural tube. When assayed one to three days later, migrating neural crest cells, motor axons, and ventral root cells were the only cells types external to the neural tube labelled with DiI. During the neural crest cell migratory phase, distinctly labelled cells were found along: (1) a dorsolateral pathway, under the epidermis, as well adjacent to and intercalating through the dermamyotome; and (2) a ventral pathway, through the rostral portion of each sclerotome and around the dorsal aorta as described previously. In contrast to those cells migrating through the sclerotome, labelled cells on the dorsolateral pathway were not segmentally arranged along the rostrocaudal axis. DiI-labelled cells were observed in all truncal neural crest derivatives, including subepidermal presumptive pigment cells, dorsal root ganglia, and sympathetic ganglia. By varying the stage at which the injection was performed, neural crest cell emigration at the level of the wing bud was shown to occur from stage 13 through stage 22. In addition, neural crest cells were found to populate their derivatives in a ventral-to-dorsal order, with the latest emigrating cells migrating exclusively along the dorsolateral pathway

Pathways of trunk neural crest cell migration in the mouse embryo as revealed by vital dye labelling

Analysis of neural crest cell migration in the mouse has been difficult due to the lack of reliable cell markers. Recently, we found that injection of DiI into the chick neural tube marks premigratory neural crest cells whose endfeet are in contact with the lumen of the neural tube (Serbedzija et al. Development 106, 809–819 (1989)). In the present study, this technique was applied to study neural crest cell migratory pathways in the trunk of the mouse embryo. Embryos were removed from the mother between the 8th and the 10th days of development and DiI was injected into the lumen of the neural tube. The embryos were then cultured for 12 to 24 h, and analyzed at the level of the forelimb. We observed two predominant pathways of neural crest cell migration: (1) a ventral pathway through the rostral portion of the somite and (2) a dorsolateral pathway between the dermamyotome and the epidermis. Neural crest cells were observed along the dorsolateral pathway throughout the period of migration. The distribution of labelled cells along the ventral pathway suggested that there were two overlapping phases of migration. An early ventrolateral phase began before E9 and ended by E9.5; this pathway consisted of a stream of cells within the rostral sclerotome, adjacent to the dermamyotome, that extended ventrally to the region of the sympathetic ganglia and the dorsal aorta

Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations

We present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level-set convex Hamilton-Jacobi equations in i.i.d. random environments, the first quantitative homogenization results for these equations in the stochastic setting. By taking advantage of a connection between the metric approach to homogenization and the theory of first-passage percolation, we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime (away from flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the flat spot), we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed, without further assumptions, at an arbitrarily slow rate. We identify a necessary and sufficient condition on the law of the Hamiltonian for an algebraic rate of convergence to hold in the sub-ballistic regime and show, under this hypothesis, that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization. Our methods are novel and quite different from the techniques employed in the periodic setting, although we benefit from previous works in both first-passage percolation and homogenization. The link between the rate of homogenization and the flat spot of the effective Hamiltonian, which is related to the nonexistence of correctors, is a purely random phenomenon observed here for the first time.Comment: 57 pages. Revised version. To appear in J. Amer. Math. So

Developmental potential of trunk neural crest cells in the mouse

The availability of naturally occurring and engineered mutations in mice which affect the neural crest makes the mouse embryo an important experimental system for studying neural crest cell differentiation. Here, we determine the normal developmental potential of neural crest cells by performing in situ cell lineage analysis in the mouse by microinjecting lysinated rhodamine dextran (LRD) into individual dorsal neural tube cells in the trunk. Labeled progeny derived from single cells were found in the neural tube, dorsal root ganglia, sympathoadrenal derivatives, presumptive Schwann cells and/or pigment cells. Most embryos contained labeled cells both in the neural tube and at least one neural crest derivative, and numerous clones contributed to multiple neural crest derivatives. The time of injection influenced the derivatives populated by the labeled cells. Injections at early stages of migration yielded labeled progeny in both dorsal and ventral neural crest derivatives, whereas those performed at later stages had labeled cells only in more dorsal neural crest derivatives, such as dorsal root ganglion and presumptive pigment cells. The results suggest that in the mouse embryo: (1) there is a common precursor for neural crest and neural tube cells; (2) some neural crest cells are multipotent; and (3) the timing of emigration influences the range of possible neural crest derivatives

Nonlinear surface impurity in a semi-infinite 2D square lattice

We examine the formation of localized states on a generalized nonlinear impurity located at, or near the surface of a semi-infinite 2D square lattice. Using the formalism of lattice Green functions, we obtain in closed form the number of bound states as well as their energies and probability profiles, for different nonlinearity parameter values and nonlinearity exponents, at different distances from the surface. We specialize to two cases: impurity close to an "edge" and impurity close to a "corner". We find that, unlike the case of a 1D semi-infinite lattice, in 2D, the presence of the surface helps the formation of a localized state.Comment: 6 pages, 7 figures, submitted to PR

Thirteen Plus One: A Comparison of Global Climate Policy Architectures

We critically review the Kyoto Protocol and thirteen alternative policy architectures for addressing the threat of global climate change. We employ six criteria to evaluate the policy proposals: environmental outcome, dynamic efficiency, cost effectiveness, equity, flexibility in the presence of new information, and incentives for participation and compliance. The Kyoto Protocol does not fare well on a number of criteria, but none of the alternative proposals fare well along all six dimensions. We identify several major themes among the alternative proposals: Kyoto is “too little, too fast”; developing countries should play a more substantial role and receive incentives to participate; implementation should focus on market-based approaches, especially those with price mechanisms; and participation and compliance incentives are inadequately addressed by most proposals. Our investigation reveals tensions among several of the evaluative criteria, such as between environmental outcome and efficiency, and between cost-effectiveness and incentives for participation and compliance.Policy architecture, Kyoto Protocol, Efficiency, Cost effectiveness, Equity, Participation, Compliance

Absolute absorption and fluorescence measurements over a dynamic range of 106^6 with cavity-enhanced laser-induced fluorescence

We describe a novel experimental setup that combines the advantages of both laser-induced fluorescence and cavity ring-down techniques. The simultaneous and correlated measurement of the ring-down and fluorescence signals yields absolute absorption coefficients for the fluorescence measurement. The combined measurement is conducted with the same sample in a single, pulsed laser beam. The fluorescence measurement extends the dynamic range of a stand-alone cavity ring-down setup from typically three to at least six orders of magnitude. The presence of the cavity improves the quality of the signal, in particular the signal-to-noise ratio. The methodology, dubbed cavity-enhanced laser-induced fluorescence (CELIF), is developed and rigorously tested against the spectroscopy of 1,4-bis(phenylethynyl)benzene in a molecular beam and density measurements in a cell. We outline how the method can be utilised to determine absolute quantities: absorption cross sections, sample densities and fluorescence quantum yields.Comment: 12 pages, 6 figures, submitted to J. Chem. Phy

Dynamical Origin of Extrasolar Planet Eccentricity Distribution

We explore the possibility that the observed eccentricity distribution of extrasolar planets arose through planet-planet interactions, after the initial stage of planet formation was complete. Our results are based on ~3250 numerical integrations of ensembles of randomly constructed planetary systems, each lasting 100 Myr. We find that for a remarkably wide range of initial conditions the eccentricity distributions of dynamically active planetary systems relax towards a common final equilibrium distribution, well described by the fitting formula dn ~ e exp[-1/2 (e/0.3)^2] de. This distribution agrees well with the observed eccentricity distribution for e > 0.2, but predicts too few planets at lower eccentricities, even when we exclude planets subject to tidal circularization. These findings suggest that a period of large-scale dynamical instability has occurred in a significant fraction of newly formed planetary systems, lasting 1--2 orders of magnitude longer than the ~1 Myr interval in which gas-giant planets are assembled. This mechanism predicts no (or weak) correlations between semimajor axis, eccentricity, inclination, and mass in dynamically relaxed planetary systems. An additional observational consequence of dynamical relaxation is a significant population of planets (>10%) that are highly inclined (>25deg) with respect to the initial symmetry plane of the protoplanetary disk; this population may be detectable in transiting planets through the Rossiter-McLaughlin effect.Comment: Accepted to ApJ, conclusions updated to reflect the current observational constraint