Approximation Theory of Multivariate Spline Functions in Sobolev Spaces

In this paper we study some approximation theory questions which arise from the analysis of the discretization error associated with the use of the Rayleigh-Ritz-Galerkin method for approximating the solutions to various types of boundary value problems, cf. [13, [2], [33, [43, [7], [8], [93, [12], [143, [18], [19], [20] and [22]. In particular, we consider upper and lower bounds for the error in approximation of certain families of functions in Sobolev spaces, cf. [15], by functions in finite-dimensional "polynomial spline types" subspaces, cf. [16]. In doing this, we directly generalize, improve, and extend the corresponding results of[1], [17], [18], [19], [20], and [21]. Throughout this paper, the symbol K will be used repeatedly to denote a positive constant, not necessarily the same at each occurrence and the symbol μ will be used repeatedly to denote a nonnegative, continuous function on [0,∞], not necessarily the same at each occurrence

L∞-Multivariate approximation theory

This paper is concerned with multivariate analogues of many standard results in the L∞-approximation theory of functions of one real variable, cf. [12], [13], [14], [20], [21] and [24]. In particular, asymptotic bounds are given for the distance of a given smooth function from various finite-dimensional spaces such as multivariate trigonometric polynomials, multivariate algebraic polynomials and multivariate polynomial spline functions

Classical Coulomb Systems:Screening and Correlations Revisited

From the laws of macroscopic electrostatics of conductors (in particular the existence of screening) taken for granted, one can deduce universal properties for the thermal fluctuations in a classical Coulomb system at equilibrium. The universality is especially apparent in the long-range correlations of the electrical potentials and fields. The charge fluctuations are derived from the field fluctuations. This is a convenient way for studying the surface charge fluctuations on a conductor with boundaries. Explicit results are given for simple geometries. The potentials and the fields have Gaussian fluctuations, except for a short-distance cutoff.Comment: 17 pages,TE

HIFI spectroscopy of low-level water transitions in M82

We present observations of the rotational ortho-water ground transition, the two lowest para-water transitions, and the ground transition of ionised ortho-water in the archetypal starburst galaxy M82, performed with the HIFI instrument on the Herschel Space Observatory. These observations are the first detections of the para-H2O(111-000) (1113\,GHz) and ortho-H2O+(111-000) (1115\,GHz) lines in an extragalactic source. All three water lines show different spectral line profiles, underlining the need for high spectral resolution in interpreting line formation processes. Using the line shape of the para-H2O(111-000) and ortho-H2O+(111-000) absorption profile in conjunction with high spatial resolution CO observations, we show that the (ionised) water absorption arises from a ~2000 pc^2 region within the HIFI beam located about ~50 pc east of the dynamical centre of the galaxy. This region does not coincide with any of the known line emission peaks that have been identified in other molecular tracers, with the exception of HCO. Our data suggest that water and ionised water within this region have high (up to 75%) area-covering factors of the underlying continuum. This indicates that water is not associated with small, dense cores within the ISM of M82 but arises from a more widespread diffuse gas component.Comment: 5 pages, 4 figures. Accepted for publication in A&

Correlations in two-component log-gas systems

A systematic study of the properties of particle and charge correlation functions in the two-dimensional Coulomb gas confined to a one-dimensional domain is undertaken. Two versions of this system are considered: one in which the positive and negative charges are constrained to alternate in sign along the line, and the other where there is no charge ordering constraint. Both systems undergo a zero-density Kosterlitz-Thouless type transition as the dimensionless coupling $\Gamma := q^2 / kT$ is varied through $\Gamma = 2$. In the charge ordered system we use a perturbation technique to establish an $O(1/r^4)$ decay of the two-body correlations in the high temperature limit. For $\Gamma \rightarrow 2^+$, the low-fugacity expansion of the asymptotic charge-charge correlation can be resummed to all orders in the fugacity. The resummation leads to the Kosterlitz renormalization equations.Comment: 39 pages, 5 figures not included, Latex, to appear J. Stat. Phys. Shortened version of abstract belo

Plant-RRBS, a bisulfite and next-generation sequencing-based methylome profiling method enriching for coverage of cytosine positions

Background: Cytosine methylation in plant genomes is important for the regulation of gene transcription and transposon activity. Genome-wide methylomes are studied upon mutation of the DNA methyltransferases, adaptation to environmental stresses or during development. However, from basic biology to breeding programs, there is a need to monitor multiple samples to determine transgenerational methylation inheritance or differential cytosine methylation. Methylome data obtained by sodium hydrogen sulfite (bisulfite)-conversion and next-generation sequencing (NGS) provide genome- wide information on cytosine methylation. However, a profiling method that detects cytosine methylation state dispersed over the genome would allow high-throughput analysis of multiple plant samples with distinct epigenetic signatures. We use specific restriction endonucleases to enrich for cytosine coverage in a bisulfite and NGS-based profiling method, which was compared to whole-genome bisulfite sequencing of the same plant material. Methods: We established an effective methylome profiling method in plants, termed plant-reduced representation bisulfite sequencing (plant-RRBS), using optimized double restriction endonuclease digestion, fragment end repair, adapter ligation, followed by bisulfite conversion, PCR amplification and NGS. We report a performant laboratory protocol and a straightforward bioinformatics data analysis pipeline for plant-RRBS, applicable for any reference-sequenced plant species. Results: As a proof of concept, methylome profiling was performed using an Oryza sativa ssp. indica pure breeding line and a derived epigenetically altered line (epiline). Plant-RRBS detects methylation levels at tens of millions of cytosine positions deduced from bisulfite conversion in multiple samples. To evaluate the method, the coverage of cytosine positions, the intra-line similarity and the differential cytosine methylation levels between the pure breeding line and the epiline were determined. Plant-RRBS reproducibly covers commonly up to one fourth of the cytosine positions in the rice genome when using MspI-DpnII within a group of five biological replicates of a line. The method predominantly detects cytosine methylation in putative promoter regions and not-annotated regions in rice. Conclusions: Plant-RRBS offers high-throughput and broad, genome- dispersed methylation detection by effective read number generation obtained from reproducibly covered genome fractions using optimized endonuclease combinations, facilitating comparative analyses of multi-sample studies for cytosine methylation and transgenerational stability in experimental material and plant breeding populations

Density Matrix Renormalisation Group Approach to the Massive Schwinger Model

The massive Schwinger model is studied, using a density matrix renormalisation group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are almost two orders of magnitude more accurate than previous calculations. Coleman's picture of `half-asymptotic' particles at background field theta = pi is confirmed. The predicted phase transition at finite fermion mass (m/g) is accurately located, and demonstrated to belong in the 2D Ising universality class.Comment: 38 pages, 18 figures, submitted to PR

HIFI spectroscopy of low-level water transitions in M82

We present observations of the rotational ortho-water ground transition, the two lowest para-water transitions, and the ground transition of ionised ortho-water in the archetypal starburst galaxy M82, performed with the HIFI instrument on the Herschel Space Observatory. These observations are the first detections of the para-H2O(111-000) (1113\,GHz) and ortho-H2O+(111-000) (1115\,GHz) lines in an extragalactic source. All three water lines show different spectral line profiles, underlining the need for high spectral resolution in interpreting line formation processes. Using the line shape of the para-H2O(111-000) and ortho-H2O+(111-000) absorption profile in conjunction with high spatial resolution CO observations, we show that the (ionised) water absorption arises from a ~2000 pc^2 region within the HIFI beam located about ~50 pc east of the dynamical centre of the galaxy. This region does not coincide with any of the known line emission peaks that have been identified in other molecular tracers, with the exception of HCO. Our data suggest that water and ionised water within this region have high (up to 75%) area-covering factors of the underlying continuum. This indicates that water is not associated with small, dense cores within the ISM of M82 but arises from a more widespread diffuse gas component.Comment: 5 pages, 4 figures. Accepted for publication in A&