Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent

The convergence of stochastic interacting particle systems in the mean-field limit to solutions to conservative stochastic partial differential equations is shown, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again with optimal rate of convergence. The results apply in particular to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions to SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of stochastic gradient descent in the continuum limit.Comment: 63 page

Coombs Hill: A Late Devonian fossil locality in the Witpoort Formation (Witteberg Group, South Africa)

Coombs Hill, a new fossil locality in the Witpoort Formation (Witteberg Group) of South Africa, preserves a record of Famennian (Late Devonian) life in Gondwana. Fossil plants collected at Coombs Hill are preliminarily assigned to several classes. Shelly invertebrates include a variety of bivalve mollusc forms, some of which appear to be preserved in life position. Biodiversity at Coombs Hill is comparable to that of the well-known Waterloo Farm lagerstätte in ordinal diversity, but exhibits differences in species composition. Ongoing taxonomic analysis will provide a rare window into the ecology of high-latitude environments during this pivotal stage of Earth history, which immediately preceded the end-Devonian extinction. Sandstone dominated sedimentary facies at Coombs Hill suggest a high-energy coastal marine setting, with brackish back-barrier estuarine/lagoonally derived fossiliferous mudstones. Exact stratigraphic placement within the Witpoort Formation is hampered by structural deformation, and precise age comparisons with Waterloo Farm are currently tenuous.Significance: A new fossil locality at Coombs Hill comprises the second known site with a suite of well preserved continental and marginal marine fossils from the Witpoort Formation, providing an exceptionally rare example of high-latitude life during the critical latest Devonian Famennian age. Several new plant taxa will be diagnosed from this locality, which also gives important insights into the morphology of Archaeopteris notosaria, South Africa’s earliest known tree. Discovery of a second palaeontologically significant site in the Witpoort Formation provides impetus for further structural and sedimentary facies analyses to align the unit with datable global eustatic events, and to clarify its internal chronology

The prevalence of visual disorders in the juvenile delinquency population

Background: Juvenile delinquent and learning disabled populations have many common characteristics. Similarities have been found in their visual systems. This study describes the visual system of a population of juvenile delinquents from Folsom, California. Methods: The records of complete analytical exams which were performed by one examiner were obtained and a data base was used to create descriptive statistics on the 477 subjects. Findings are compared to OEP Expected Values, Morgan\u27s Normative Values, case analysis, and current studies in the literature regarding the visual profile of juvenile delinquents and the learning disabled. Results: The study revealed that the Folsom, California juvenile delinquent population has significantly decreased accommodation and vergence skills at near when compared with OEP Expected Values and Morgan\u27s Normative Values. They are a highly symptomatic group complaining of blur, headaches, and losing place while reading. Visual perception and ocular motility problems also predominate. Conclusions: There appears to be a significant need for plus nearpoint lenses and vision training for visual perception and ocular motilities among the juvenile delinquent population

Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads

We use a theoretical model to explore how fluid dynamics, in particular, the pressure gradient and wall shear stress in a channel, affect the deposition of particles flowing in a microfluidic network. Experiments on transport of colloidal particles in pressure-driven systems of packed beads have shown that at lower pressure drop, particles deposit locally at the inlet, while at higher pressure drop, they deposit uniformly along the direction of flow. We develop a mathematical model and use agent-based simulations to capture these essential qualitative features observed in experiments. We explore the deposition profile over a two-dimensional phase diagram defined in terms of the pressure and shear stress threshold, and show that two distinct phases exist. We explain this apparent phase transition by drawing an analogy to simple one-dimensional models of aggregation in which the phase transition is calculated analytically.Comment: 9 pages, 6 figures including Supplemental Materia

Random attractors for degenerate stochastic partial differential equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate p-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate p-Laplace equations we prove that the deterministic, infinite dimensional attractor collapses to a single random point if enough noise is added.Comment: 34 pages; The final publication is available at http://link.springer.com/article/10.1007%2Fs10884-013-9294-

From teaching physics to teaching children : beginning teachers learning from pupils

This paper discusses the development of beginning physics teachers' pedagogical content knowledge (PCK) in the context of teaching basic electricity during a one-year Professional Graduate Diploma in Education course (PGDE) and beyond. This longitudinal study used repeated semi-structured interviews over a period of four-and-a-half years. The interview schedule followed a line of development through the secondary school electrical syllabus in Scotland. Fifteen student teachers were interviewed during the PGDE year. Six of them were followed up at the end of the Induction Year (their first year as a newly qualified teacher), and again two-and-a-half years later. Thematic analysis of the interviews showed that before the beginning teachers had taught any classes, their initial focus was on how to transform their own subject matter knowledge (SMK) about electricity into forms that were accessible to pupils. As the beginning teachers gained experience working with classes, they gave vivid descriptions of interacting with particular pupils when teaching electricity which showed the development of their pedagogical knowledge. This played a significant role in the teachers' change of focus from teaching physics to teaching children as they transformed their SMK into forms that were accessible to pupils and developed their general pedagogical knowledge

New morphological information on, and species of placoderm fish Africanaspis (Arthrodira, Placodermi) from the Late Devonian of South Africa

Here we present a new species of placoderm fish, Africanaspis edmountaini sp. nov., and redescribe Africanaspis doryssa on the basis of new material collected from the type locality of Africanaspis. The new material includes the first head shields of Africanaspis doryssa in addition to soft anatomy for both taxa. Hitherto Africanaspis was entirely described from trunk armour and no record of body and fin outlines had been recorded. In addition the first record of embryonic and juvenile specimens of Africanaspis doryssa is presented and provides a growth series from presumed hatchlings to presumed adults. The presence of a greater number of juveniles compared to adults indicates that the Waterloo Farm fossil site in South Africa represents the first nursery site of arthrodire placoderms known from a cold water environment. The preservation of an ontogenetic series demonstrates that variation within the earlier known sample, initially considered to have resulted from ontogenetic change, instead indicates the presence of a second, less common species Africanaspis edmountaini sp. nov. There is some faunal overlap between the Waterloo Farm fossil site and faunas described from Strud in Belgium and Red Hill, Pennsylvania, in north America, supporting the concept of a more cosmopolitan vertebrate fauna in the Famennian than earlier in the Devonian

Sodium-Dependent Vitamin C Transporter 2 (SVCT2) Expression and Activity in Brain Capillary Endothelial Cells after Transient Ischemia in Mice

Expression and transport activity of Sodium-dependent Vitamin C Transporter 2 (SVCT2) was shown in various tissues and organs. Vitamin C was shown to be cerebroprotective in several animal models of stroke. Data on expression, localization and transport activity of SVCT2 after cerebral ischemia, however, has been scarce so far. Thus, we studied the expression of SVCT2 after middle cerebral artery occlusion (MCAO) in mice by immunohistochemistry. We found an upregulation of SVCT2 after stroke. Co-stainings with Occludin, Von-Willebrand Factor and CD34 demonstrated localization of SVCT2 in brain capillary endothelial cells in the ischemic area after stroke. Time-course analyses of SVCT2 expression by immunohistochemistry and western blots showed upregulation in the subacute phase of 2–5 days. Radioactive uptake assays using 14C-labelled ascorbic acid showed a significant increase of ascorbic acid uptake into the brain after stroke. Taken together, these results provide evidence for the expression and transport activity of SVCT2 in brain capillary endothelial cells after transient ischemia in mice. These results may lead to the development of novel neuroprotective strategies in stroke therapy