An improved sum-product estimate for general finite fields

An improved sum-product estimate for subsets of a finite field whose order is not prime is provided. It is shown, under certain conditions, that max{∣∣∣A+A∣∣∣,∣∣∣A⋅A∣∣∣}≫∣∣A∣∣12/11(log2∣∣A∣∣)5/11. This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

A nonparametric ensemble transform method for Bayesian inference

Many applications, such as intermittent data assimilation, lead to a recursive application of Bayesian inference within a Monte Carlo context. Popular data assimilation algorithms include sequential Monte Carlo methods and ensemble Kalman filters (EnKFs). These methods differ in the way Bayesian inference is implemented. Sequential Monte Carlo methods rely on importance sampling combined with a resampling step, while EnKFs utilize a linear transformation of Monte Carlo samples based on the classic Kalman filter. While EnKFs have proven to be quite robust even for small ensemble sizes, they are not consistent since their derivation relies on a linear regression ansatz. In this paper, we propose another transform method, which does not rely on any a priori assumptions on the underlying prior and posterior distributions. The new method is based on solving an optimal transportation problem for discrete random variables. © 2013, Society for Industrial and Applied Mathematics

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.

A finite element method for second order nonvariational elliptic problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Assessing the quality of modelled 3D protein structures using the ModFOLD server

Model quality assessment programs (MQAPs) aim to assess the quality of modelled 3D protein structures. The provision of quality scores, describing both global and local (per-residue) accuracy are extremely important, as without quality scores we are unable to determine the usefulness of a 3D model for further computational and experimental wet lab studies.Here, we briefly discuss protein tertiary structure prediction, along with the biennial Critical Assessment of Techniques for Protein Structure Prediction (CASP) competition and their key role in driving the field of protein model quality assessment methods (MQAPs). We also briefly discuss the top MQAPs from the previous CASP competitions. Additionally, we describe our downloadable and webserver-based model quality assessment methods: ModFOLD3, ModFOLDclust, ModFOLDclustQ, ModFOLDclust2, and IntFOLD-QA. We provide a practical step-by-step guide on using our downloadable and webserver-based tools and include examples of their application for improving tertiary structure prediction, ligand binding site residue prediction, and oligomer predictions.

Prediction of Staphylococcus aureus antimicrobial resistance by whole-genome sequencing

Whole-genome sequencing (WGS) could potentially provide a single platform for extracting all the information required to predict an organism’s phenotype. However, its ability to provide accurate predictions has not yet been demonstrated in large independent studies of specific organisms. In this study, we aimed to develop a genotypic prediction method for antimicrobial susceptibilities. The whole genomes of 501 unrelated Staphylococcus aureus isolates were sequenced, and the assembled genomes were interrogated using BLASTn for a panel of known resistance determinants (chromosomal mutations and genes carried on plasmids). Results were compared with phenotypic susceptibility testing for 12 commonly used antimicrobial agents (penicillin, methicillin, erythromycin, clindamycin, tetracycline, ciprofloxacin, vancomycin, trimethoprim, gentamicin, fusidic acid, rifampin, and mupirocin) performed by the routine clinical laboratory. We investigated discrepancies by repeat susceptibility testing and manual inspection of the sequences and used this information to optimize the resistance determinant panel and BLASTn algorithm. We then tested performance of the optimized tool in an independent validation set of 491 unrelated isolates, with phenotypic results obtained in duplicate by automated broth dilution (BD Phoenix) and disc diffusion. In the validation set, the overall sensitivity and specificity of the genomic prediction method were 0.97 (95% confidence interval [95% CI], 0.95 to 0.98) and 0.99 (95% CI, 0.99 to 1), respectively, compared to standard susceptibility testing methods. The very major error rate was 0.5%, and the major error rate was 0.7%. WGS was as sensitive and specific as routine antimicrobial susceptibility testing methods. WGS is a promising alternative to culture methods for resistance prediction in S. aureus and ultimately other major bacterial pathogens.

Analysis of the bacterial communities present in lungs of patients with cystic fibrosis from American and British centers

The aim of this study was to determine whether geographical differences impact the composition of bacterial communities present in the airways of cystic fibrosis (CF) patients attending CF centers in the United States or United Kingdom. Thirty-eight patients were matched on the basis of clinical parameters into 19 pairs comprised of one U.S. and one United Kingdom patient. Analysis was performed to determine what, if any, bacterial correlates could be identified. Two culture-independent strategies were used: terminal restriction fragment length polymorphism (T-RFLP) profiling and 16S rRNA clone sequencing. Overall, 73 different terminal restriction fragment lengths were detected, ranging from 2 to 10 for U.S. and 2 to 15 for United Kingdom patients. The statistical analysis of T-RFLP data indicated that patient pairing was successful and revealed substantial transatlantic similarities in the bacterial communities. A small number of bands was present in the vast majority of patients in both locations, indicating that these are species common to the CF lung. Clone sequence analysis also revealed that a number of species not traditionally associated with the CF lung were present in both sample groups. The species number per sample was similar, but differences in species presence were observed between sample groups. Cluster analysis revealed geographical differences in bacterial presence and relative species abundance. Overall, the U.S. samples showed tighter clustering with each other compared to that of United Kingdom samples, which may reflect the lower diversity detected in the U.S. sample group. The impact of cross-infection and biogeography is considered, and the implications for treating CF lung infections also are discussed.

The dynamics of Baroclinic Zonal jets

Multiple alternating zonal jets are a ubiquitous feature of planetary atmospheres and oceans. However, most studies to date have focused on the special case of barotropic jets. Here, the dynamics of freely evolving baroclinic jets are investigated using a two-layer quasigeostrophic annulus model with sloping topography. In a suite of 15 numerical simulations, the baroclinic Rossby radius and baroclinic Rhines scale are sampled by varying the stratification and root-mean-square eddy velocity, respectively. Small-scale eddies in the initial state evolve through geostrophic turbulence and accelerate zonally as they grow in horizontal scale, first isotropically and then anisotropically. This process leads ultimately to the formation of jets, which take about 2500 rotation periods to equilibrate. The kinetic energy spectrum of the equilibrated baroclinic zonal flow steepens from a −3 power law at small scales to a −5 power law near the jet scale. The conditions most favorable for producing multiple alternating baroclinic jets are large baroclinic Rossby radius (i.e., strong stratification) and small baroclinic Rhines scale (i.e., weak root-mean-square eddy velocity). The baroclinic jet width is diagnosed objectively and found to be 2.2–2.8 times larger than the baroclinic Rhines scale, with a best estimate of 2.5 times larger. This finding suggests that Rossby wave motions must be moving at speeds of approximately 6 times the turbulent eddy velocity in order to be capable of arresting the isotropic inverse energy cascade.

Global networks, cities and economic performance: Observations from an analysis of cities in Europe and the USA

The network paradigm has been highly influential in spatial analysis in the globalisation era. As economies across the world have become increasingly integrated, so-called global cities have come to play a growing role as central nodes in the networked global economy. The idea that a city’s position in global networks benefits its economic performance has resulted in a competitive policy focus on promoting the economic growth of cities by improving their network connectivity. However, in spite of the attention being given to boosting city connectivity little is known about whether this directly translates to improved city economic performance and, if so, how well connected a city needs to be in order to benefit from this. In this paper we test the relationship between network connectivity and economic performance between 2000 and 2008 for cities with over 500,000 inhabitants in Europe and the USA to inform European policy.

A Posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

Na5(Ga4S)(GaS4)3·6H2O: a three-dimensional thiogallate containing a novel octahedral building block

The new thiogallate Na5(Ga4S)(GaS4)3·6H2O has been prepared solvothermally, using 3,5-dimethyl pyridine as a solvent, and characterised by powder and single crystal X-ray diffraction. This material, which exhibits a three-dimensional crystal structure, crystallises in the cubic space group View the MathML sourceF4¯3c (a = 17.557(4) Å). The crystal structure contains octahedral building blocks [Ga4S (GaS4)6]20−, linked into a three-dimensional network with a perovskite-type topology, and sodium hydrate clusters, [Na5(H2O)6]5+, filling the cavities in the [Ga4S(GaS4)6/2]5− framework. UV–Vis diffuse reflectance measurements indicate that this material is a wide band gap semiconductor, with a band gap of ca. 4.4 eV.