Representation of the Lagrange reconstructing polynomial by combination of substencils

The Lagrange reconstructing polynomial [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] of a function $f(x)$ on a given set of equidistant (\Delta x=\const) points $\bigl\{x_i+\ell\Delta x;\;\ell\in\{-M_-,...,+M_+\}\bigr\}$ is defined [Gerolymos G.A.: {\em J. Approx. Theory} {\bf 163} (2011) 267--305] as the polynomial whose sliding (with $x$) averages on $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to the Lagrange interpolating polynomial of $f(x)$ on the same stencil. We first study the fundamental functions of Lagrange reconstruction, show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) $x_i+n\tfrac{1}{2}\Delta x$ ($n\in\mathbb{Z}$), and obtain several identities. Using these identities, by analogy to the recursive Neville-Aitken-like algorithm applied to the Lagrange interpolating polynomial, we show that there exists a unique representation of the Lagrange reconstructing polynomial on $\{i-M_-,...,i+M_+\}$ as a combination of the Lagrange reconstructing polynomials on the $K_\mathrm{s}+1\leq M:=M_-+M_+>1$ substencils $\{i-M_-+k_\mathrm{s},...,i+M_+-K_\mathrm{s}+k_\mathrm{s}\}$ ($k_\mathrm{s}\in\{0,...,K_\mathrm{s}\}$), with weights $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ which are rational functions of $\xi$ ($x=x_i+\xi\Delta x$) [Liu Y.Y., Shu C.W., Zhang M.P.: {\em Acta Math. Appl. Sinica} {\bf 25} (2009) 503--538], and give an analytical recursive expression of the weight-functions. We then use the analytical expression of the weight-functions $\sigma_{R_1,M_-,M_+,K_\mathrm{s},k_\mathrm{s}}(\xi)$ to obtain a formal proof of convexity (positivity of the weight-functions) in the neighborhood of $\xi=\tfrac{1}{2}$, under the condition that all of the substencils contain either point $i$ or point $i+1$ (or both).Comment: final corrected version; in print J. Comp. Appl. Mat

Biodiversity and ecosystem function in soil

1. Soils are one of the last great frontiers for biodiversity research and are home to an extraordinary range of microbial and animal groups. Biological activities in soils drive many of the key ecosystem processes that govern the global system, especially in the cycling of elements such as carbon, nitrogen and phosphorus. 2. We cannot currently make firm statements about the scale of biodiversity in soils, or about the roles played by soil organisms in the transformations of organic materials that underlie those cycles. The recent UK Soil Biodiversity Programme (SBP) has brought a unique concentration of researchers to bear on a single soil in Scotland, and has generated a large amount of data concerning biodiversity, carbon flux and resilience in the soil ecosystem. 3. One of the key discoveries of the SBP was the extreme diversity of small organisms: researchers in the programme identified over 100 species of bacteria, 350 protozoa, 140 nematodes and 24 distinct types of arbuscular mycorrhizal fungi. Statistical analysis of these results suggests a much greater 'hidden diversity'. In contrast, there was no unusual richness in other organisms, such as higher fungi, mites, collembola and annelids. 4. Stable-isotope (C-13) technology was used to measure carbon fluxes and map the path of carbon through the food web. A novel finding was the rapidity with which carbon moves through the soil biota, revealing an extraordinarily dynamic soil ecosystem. 5. The combination of taxonomic diversity and rapid carbon flux makes the soil ecosystem highly resistant to perturbation through either changing soil structure or removing selected groups of organisms

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney–Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein–Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper

Directional point-contact spectroscopy of MgB2 single crystals in magnetic fields: two-band superconductivity and critical fields

The results of the first directional point-contact measurements in MgB2 single crystals, in the presence of magnetic fields up to 9 T either parallel or perpendicular to the ab planes, are presented. By applying suitable magnetic fields, we separated the partial contributions of the sigma and pi bands to the total Andreev-reflection conductance. Their fit with the BTK model allowed a very accurate determination of the temperature dependency of the gaps (Delta_sigma and Delta_pi), that resulted in close agreement with the predictions of the two-band models for MgB2. We also obtained, for the first time with point-contact spectroscopy, the temperature dependence of the (anisotropic) upper critical field of the sigma band and of the (isotropic) upper critical field of the pi band.Comment: 2 pages, 2 figures, proceedings of M2S-HTSC-VII conference, Rio de Janeiro (May 2003

Massless BTZ black holes in minisuperspace

We study aspects of the propagation of strings on BTZ black holes. After performing a careful analysis of the global spacetime structure of generic BTZ black holes, and its relation to the geometry of the SL(2,R) group manifold, we focus on the simplest case of the massless BTZ black hole. We study the SL(2,R) Wess-Zumino-Witten model in the worldsheet minisuperspace limit, taking into account special features associated to the Lorentzian signature of spacetime. We analyse the two- and three-point functions in the pointparticle limit. To lay bare the underlying group structure of the correlation functions, we derive new results on Clebsch-Gordan coefficients for SL(2,R) in a parabolic basis. We comment on the application of our results to string theory in singular time-dependent orbifolds, and to a Lorentzian version of the AdS/CFT correspondence.Comment: 28 pages, v2: reference adde

An Architecture for Data and Knowledge Acquisition for the Semantic Web: the AGROVOC Use Case

We are surrounded by ever growing volumes of unstructured and weakly-structured information, and for a human being, domain expert or not, it is nearly impossible to read, understand and categorize such information in a fair amount of time. Moreover, different user categories have different expectations: final users need easy-to-use tools and services for specific tasks, knowledge engineers require robust tools for knowledge acquisition, knowledge categorization and semantic resources development, while semantic applications developers demand for flexible frameworks for fast and easy, standardized development of complex applications. This work represents an experience report on the use of the CODA framework for rapid prototyping and deployment of knowledge acquisition systems for RDF. The system integrates independent NLP tools and custom libraries complying with UIMA standards. For our experiment a document set has been processed to populate the AGROVOC thesaurus with two new relationships

Approximation error of the Lagrange reconstructing polynomial

The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor

Surface critical behavior in fixed dimensions d<4d<4: Nonanalyticity of critical surface enhancement and massive field theory approach

The critical behavior of semi-infinite systems in fixed dimensions $d<4$ is investigated theoretically. The appropriate extension of Parisi's massive field theory approach is presented.Two-loop calculations and subsequent Pad\'e-Borel analyses of surface critical exponents of the special and ordinary phase transitions yield estimates in reasonable agreement with recent Monte Carlo results. This includes the crossover exponent $\Phi (d=3)$, for which we obtain the values $\Phi (n=1)\simeq 0.54$ and $\Phi (n=0)\simeq 0.52$, considerably lower than the previous $\epsilon$-expansion estimates.Comment: Latex with Revtex-Stylefiles, 4 page

Bonding in MgSi and AlMgSi Compounds Relevant to AlMgSi Alloys

The bonding and stability of MgSi and AlMgSi compounds relevant to AlMgSi alloys is investigated with the use of (L)APW+(lo) DFT calculations. We show that the $\beta$ and $\beta''$ phases found in the precipitation sequence are characterised by the presence of covalent bonds between Si-Si nearest neighbour pairs and covalent/ionic bonds between Mg-Si nearest neighbour pairs. We then investigate the stability of two recently discovered precipitate phases, U1 and U2, both containing Al in addition to Mg and Si. We show that both phases are characterised by tightly bound Al-Si networks, made possible by a transfer of charge from the Mg atoms.Comment: 11 pages, 30 figures, submitted to Phys. Rev.