The Global Dynamics of Discrete Semilinear Parabolic Equations

A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty$. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied

Hornblendite delineates zones of mass transfer through the lower crust

Geochemical signatures throughout the layered Earth require significant mass transfer through the lower crust, yet geological pathways are under-recognized. Elongate bodies of basic to ultrabasic rocks are ubiquitous in exposures of the lower crust. Ultrabasic hornblendite bodies hosted within granulite facies gabbroic gneiss of the Pembroke Valley, Fiordland, New Zealand, are typical occurrences usually reported as igneous cumulate hornblendite. Their igneous features contrast with the metamorphic character of their host gabbroic gneiss. Both rock types have a common parent; field relationships are consistent with modification of host gabbroic gneiss into hornblendite. This precludes any interpretation involving cumulate processes in forming the hornblendite; these bodies are imposter cumulates. Instead, replacement of the host gabbroic gneiss formed hornblendite as a result of channeled high melt flux through the lower crust. High melt/rock ratios and disequilibrium between the migrating magma (granodiorite) and its host gabbroic gneiss induced dissolution (grain-scale magmatic assimilation) of gneiss and crystallization of mainly hornblende from the migrating magma. The extent of this reaction-replacement mechanism indicates that such hornblendite bodies delineate significant melt conduits. Accordingly, many of the ubiquitous basic to ultrabasic elongate bodies of the lower crust likely map the ‘missing’ mass transfer zones

Reconciling Bayesian and Total Variation Methods for Binary Inversion

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization. On the other hand, sparse or noisy data often demands a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion is a natural framework in which to carry this out. The link between Bayesian inversion methods and perimeter regularization, however, is not fully understood. In this paper two links are studied: (i) the MAP objective function of a suitably chosen phase-field Bayesian approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess finite perimeter and to have the ability to learn about the true perimeter. Furthermore, the level set approach is shown to lead to faster algorithms for uncertainty quantification than the phase field approach

The viscous Cahn-Hilliard equation. I. Computations

The viscous Cahn-Hilliard equation arises as a singular limit of the phase-field model of phase transitions. It contains both the Cahn-Hilliard and Allen-Cahn equations as particular limits. The equation is in gradient form and possesses a compact global attractor A, comprising heteroclinic orbits between equilibria. Two classes of computation are described. First heteroclinic orbits on the global attractor are computed; by using the viscous Cahn-Hilliard equation to perform a homotopy, these results show that the orbits, and hence the geometry of the attractors, are remarkably insensitive to whether the Allen-Cahn or Cahn-Hilliard equation is studied. Second, initial-value computations are described; these computations emphasize three differing mechanisms by which interfaces in the equation propagate for the case of very small penalization of interfacial energy. Furthermore, convergence to an appropriate free boundary problem is demonstrated numerically

Doubly Robust Inference when Combining Probability and Non-probability Samples with High-dimensional Data

Non-probability samples become increasingly popular in survey statistics but may suffer from selection biases that limit the generalizability of results to the target population. We consider integrating a non-probability sample with a probability sample which provides high-dimensional representative covariate information of the target population. We propose a two-step approach for variable selection and finite population inference. In the first step, we use penalized estimating equations with folded-concave penalties to select important variables for the sampling score of selection into the non-probability sample and the outcome model. We show that the penalized estimating equation approach enjoys the selection consistency property for general probability samples. The major technical hurdle is due to the possible dependence of the sample under the finite population framework. To overcome this challenge, we construct martingales which enable us to apply Bernstein concentration inequality for martingales. In the second step, we focus on a doubly robust estimator of the finite population mean and re-estimate the nuisance model parameters by minimizing the asymptotic squared bias of the doubly robust estimator. This estimating strategy mitigates the possible first-step selection error and renders the doubly robust estimator root-n consistent if either the sampling probability or the outcome model is correctly specified

Labor-associated gene expression in the human uterine fundus, lower segment, and cervix

Background Preterm labor, failure to progress, and postpartum hemorrhage are the common causes of maternal and neonatal mortality or morbidity. All result from defects in the complex mechanisms controlling labor, which coordinate changes in the uterine fundus, lower segment, and cervix. We aimed to assess labor-associated gene expression profiles in these functionally distinct areas of the human uterus by using microarrays. Methods and Findings Samples of uterine fundus, lower segment, and cervix were obtained from patients at term (mean +/- 6 SD = 39.1 +/- 0.5 wk) prior to the onset of labor (n = 6), or in active phase of labor with spontaneous onset (n = 7). Expression of 12,626 genes was evaluated using microarrays ( Human Genome U95A; Affymetrix) and compared between labor and non-labor samples. Genes with the largest labor-associated change and the lowest variability in expression are likely to be fundamental for parturition, so gene expression was ranked accordingly. From 500 genes with the highest rank we identified genes with similar expression profiles using two independent clustering techniques. Sets of genes with a probability of chance grouping by both techniques less than 0.01 represented 71.2%, 81.8%, and 79.8% of the 500 genes in the fundus, lower segment, and cervix, respectively. We identified 14, 14, and 12 those sets of genes in the fundus, lower segment, and cervix, respectively. This enabled networks of coregulated and co-expressed genes to be discovered. Many genes within the same cluster shared similar functions or had functions pertinent to the process of labor. Conclusions Our results provide support for many of the established processes of parturition and also describe novel-to-labor genes not previously associated with this process. The elucidation of these mechanisms likely to be fundamental for controlling labor is an important prerequisite to the development of effective treatments for major obstetric problems - including prematurity, with its long-term consequences to the health of mother and offspring

Reconciling Bayesian and Perimeter Regularization for Binary Inversion

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of total variation. On the other hand, sparse or noisy data often demand a probabilistic approach to the reconstruction of images, to enable uncertainty quantification; the Bayesian approach to inversion, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori objective function of a suitably chosen Bayesian phase-field approach is shown to be closely related to a least squares plus perimeter regularization objective; (ii) sample paths of a suitably chosen Bayesian level set formulation are shown to possess a finite perimeter and to have the ability to learn about the true perimeter

Mechanisms of high-frequency song generation in brachypterous crickets and the role of ghost frequencies

Sound production in crickets relies on stridulation, the well-understood rubbing together of a pair of specialised wings. As the file of one wing slides over the scraper of the other, a series of rhythmic impacts cause harmonic oscillations, usually resulting in the radiation of pure tones delivered at low frequencies (2-8 kHz). In the short winged crickets of the Lebinthini tribe, acoustic communication relies on signals with remarkably high frequencies (> 8 kHz) and rich harmonic content. Using several species of the subfamily Eneopterinae, we characterise the morphological and mechanical specialisations supporting the production of high frequencies, and demonstrate that higher harmonics are exploited as dominant frequencies. These specialisations affect the structure of the stridulatory file, the motor control of stridulation and the resonance of the sound radiator. We place these specialisations in a phylogenetic framework and show that they serve to exploit high frequency vibrational modes pre-existing in the phylogenetic ancestor. In Eneopterinae, the lower frequency components are harmonically related to the dominant peak, suggesting they are relicts of ancestral carrier frequencies. Yet, such ghost frequencies still occur in the wings' free resonances, highlighting the fundamental mechanical constraints of sound radiation. These results support the hypothesis that such high frequency songs evolved stepwise, by a form of punctuated evolution which could be related to functional constraints, rather than by the progressive increase of the ancestral fundamental frequency

Perturbations of nuclear C*-algebras

Kadison and Kastler introduced a natural metric on the collection of all C*-subalgebras of the bounded operators on a separable Hilbert space. They conjectured that sufficiently close algebras are unitarily conjugate. We establish this conjecture when one algebra is separable and nuclear. We also consider one-sided versions of these notions, and we obtain embeddings from certain near inclusions involving separable nuclear C*-algebras. At the end of the paper we demonstrate how our methods lead to improved characterisations of some of the types of algebras that are of current interest in the classification programme.Comment: 45 page