Some non-linear s.p.d.e.'s that are second order in time

We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms

Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hitting properties of parabolic s.p.d.e.'s with reflection

We study the hitting properties of the solutions $u$ of a class of parabolic stochastic partial differential equations with singular drifts that prevent $u$ from becoming negative. The drifts can be a reflecting term or a nonlinearity $cu^{-3}$, with $c>0$. We prove that almost surely, for all time $t>0$, the solution $u_t$ hits the level 0 only at a finite number of space points, which depends explicitly on $c$. In particular, this number of hits never exceeds 4 and if $c>15/8$, then level 0 is not hit.Comment: Published at http://dx.doi.org/10.1214/009117905000000792 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs

Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure correction has to be solved at every time step in the dynamical core of many numerical weather prediction models, and equations of a very similar structure arise in global ocean models, subsurface flow simulations and gas and oil reservoir modelling. The elliptic solve is often the bottleneck of the forecast, and an algorithmically optimal method has to be used and implemented efficiently. Graphics Processing Units have been shown to be highly efficient for a wide range of applications in scientific computing, and recently iterative solvers have been parallelised on these architectures. We describe the GPU implementation and optimisation of a Preconditioned Conjugate Gradient (PCG) algorithm for the solution of a three dimensional anisotropic elliptic PDE for the pressure correction in NWP. Our implementation exploits the strong vertical anisotropy of the elliptic operator in the construction of a suitable preconditioner. As the algorithm is memory bound, performance can be improved significantly by reducing the amount of global memory access. We achieve this by using a matrix-free implementation which does not require explicit storage of the matrix and instead recalculates the local stencil. Global memory access can also be reduced by rewriting the algorithm using loop fusion and we show that this further reduces the runtime on the GPU. We demonstrate the performance of our matrix-free GPU code by comparing it to a sequential CPU implementation and to a matrix-explicit GPU code which uses existing libraries. The absolute performance of the algorithm for different problem sizes is quantified in terms of floating point throughput and global memory bandwidth.Comment: 18 pages, 7 figure

Global banking and international business cycles

This paper incorporates a global bank into a two-country business-cycle model. The bank collects deposits from households and makes loans to entrepreneurs, in both countries. It has to finance a fraction of loans using equity. We investigate how such a bank capital requirement affects the international transmission of productivity and loan default shocks. Three findings emerge. First, the bank's capital requirement has little effect on the international transmission of productivity shocks. Second, the contribution of loan default shocks to business cycle fluctuations is negligible under normal economic conditions. Third, an exceptionally large loan loss originating in one country induces a sizeable and simultaneous decline in economic activity in both countries. This is particularly noteworthy, as the 2007–09 global financial crisis was characterized by large credit losses in the US and a simultaneous sharp output reduction in the U.S. and the euro Area. Our results thus suggest that global banks may have played an important role in the international transmission of the crisis.Equity ; Bank capital ; Productivity ; Default (Finance) ; Loans

SGN Database: From QTLs to Genomes

Quantitative trait loci (QTL) analysis is used to dissect the genetic basis underlying polygenic traits. Several public databases have been storing and making QTL data available to research communities. To our knowledge, current QTL databases rely on manual curation where curators read literature and extract relevant QTL information to store in databases. Evidently, this approach is expensive in terms of expert manpower and time use and limits the type of data that can be curated. At the Solanaceae Genomics Network (SGN) (&#x22;http://sgn.cornell.edu&#x22;:http://sgn.cornell.edu), we have developed a database to store raw phenotype and genotype data from QTL studies, perform, on the fly, QTL analysis using R/QTL statistical software (&#x22;http://www.rqtl.org&#x22;:http://www.rqtl.org) and visualize QTLs on a genetic map. Users can identify peak, and flanking markers for QTLs of traits of interest. The QTL database is integrated with other SGN databases (eg. Marker, BACs, and Unigenes), and analysis tools such as the Comparative Map Viewer. Using the comparative map viewer, users can compare chromosome with QTL regions to genetic maps of interest from the same or different Solanaceae species. As the tomato genome sequencing advances, users can also identify corresponding BAC sequences or locations on the tomato physical map, which can be suggestive of candidate genes for a trait of interest.&#xd;&#xa;&#xd;&#xa;Furthermore at SGN, images, quantitative phenotype and genotype data, publications, genetic maps generated by QTL studies are displayed and available for download. Currently, data from three F2 and two backcross population QTL studies on fruit morphology traits (18 &#x2013; 46 traits per population) is available at the SGN website for viewing at population, accession, and trait levels. Traits are described using ontology terms. Phenotype data is presented in tabular and graphical formats such as frequency distributions with basic descriptive statistics. Mapping data showing location of parental alleles on individual accession genetic maps is also available.&#xd;&#xa;&#xd;&#xa;SGN is a public database hosted at Boyce Thomson Institute, Cornell University, and funded by USDA CSREES and NSF

Characterisation of fungal symbionts and microbial communities of Austroplatypus incompertus (Platypodinae) and other Australian ambrosia beetle species

Insects are the most diverse taxonomic class and live often in a tight symbiotic associations with a microbial cosmos that is expected to be even more diverse. One of these symbioses is fungal farming, the ability to propagate, cultivate and harvest fungi as a primary food source. This ability evolved independently in three major insect orders, attine ants, macrotermitine termites and ambrosia beetles. While many studies have been conducted on the former two orders, the diversity and mechanisms of fungal farming in ambrosia beetles are less understood. Ambrosia beetles include two subfamilies, Scolytinae (bark beetles) and Platypodinae (pinhole borers). Platypodinae species are almost all fungal famers evolved roughly 90 million years ago, and fungal farming is likely to have evolved once in this lineage. Therefore, they possibly constitute the oldest lineage of fungal farming insects, while fungal farming in Scolytinae evolved multiple times and is much younger (~ 50 million years). There are 1,400 species of described Platypodinae species, however less than 1% of their fungal associates have been described. The diversity lurking in tropical biogeographic regions of South America, Asia and Australia is expected to be large, and this, therefore, creates a great opportunity to discover and describe new species in these regions. The aim of this thesis was to contribute to a better understanding of the fungal farming in Platypodinae by investigating the microbial community of four Australian platypodine species. So far 46 Platypodinae species have been recorded for this continent, however none of their fungal or bacterial partners have been formally described. This thesis is the first systematic analysis and characterisation of the fungal symbionts and the microbiome of Australian Platypodinae species. It provides a first insight in the microbial community diversity and mechanisms shaping these communities in two ancient and two more derived ambrosia beetle species, and therefore constitutes an important contribution to the study of the ecology and evolution of ambrosia beetles