Some extremal functions in Fourier analysis, II

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erd\"{o}s-Tur\'{a}n-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.Comment: 40 pages. Accepted for publication in Trans. Amer. Math. So

Gaussian Subordination for the Beurling-Selberg Extremal Problem

We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and \cite{Lit}), plus a variety of new interesting functions such as $|x|^{\alpha}$ for $-1 < \alpha$; \,$\log \,\bigl((x^2 + \alpha^2)/(x^2 + \beta^2)\bigr)$, for $0 \leq \alpha < \beta$;\, $\log\bigl(x^2 + \alpha^2\bigr)$; and $x^{2n} \log x^2$\,, for $n \in \N$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one

The Unsettled Church: The Search for Identity and Relevance in the Ecclesiologies of Nicholas Healy, Ephraim Radner, and Darrell Guder

This dissertation examines the efforts of three contemporary theologians whose work is a part of the search for a new methodology for doing ecclesiology located on the continuum between the Church\u27s identity and relevance. They are the Catholic theologian Nicholas Healy, Anglican theologian Ephraim Radner, and Presbyterian theologian Darrell Guder. They come to the subject matter from different ecclesiological backgrounds, and, as such, their work can be taken as representative in as much as it stands for their unique efforts to theologize within their own traditions and contexts. By critiquing and analyzing their proposals I will bring them into dialog which will yield what I hope are the contours of a new way of thinking about ecclesiology. In my study of their ecclesiological proposals I examine first their approach to the Holy Scriptures relative to their search for the Church\u27s identity. Special attention will be given to Christological and Pneumatological concerns, but also to the role of corporate and individual repentance (or conversion), as a means of re-appropriating one\u27s true identity as Church. Second, I take a critical look at their proposals of how the Church\u27s identity can and should enable its practical embodiment in the context of the 21st Century\u27s marketplace of ideas and be expressed in its God-given mission, i.e., the Church\u27s relevance. Then, based on a careful examination of the postmodern context, I argue that the aforementioned theologians represent the emergence of a new methodological axis, namely the one defined by identity and relevance, for doing ecclesiology. I will argue that this methodological axis gives rise to a new model, which I will call, missional ecclesiology. My critical evaluation of this new methodology concludes with an evaluation of its potential viability arguing in favor of missional ecclesiology as a viable model

Imaging the Alpine crust with ambient-noise tomography: linking surface observations to deep structures

By making use of the data coverage from the AlpArray and Swath D networks, a dense set of ambient-noise measurements of Rayleigh and Love waves is extracted. This data is used to investigate the Alpine crustal and uppermost mantle structure using different approaches: (1) Azimuthal anisotropy from Eikonal tomography for the AlpArray network. We show how Eikonal tomography can be used to extract azimuthal anisotropy from surface-wave data. The methodological advantages and difficulties are discussed in detail. It is found that strong velocity heterogeneities can be the source of a major bias by causing strongly deformed wavefronts. By averaging contributions from many azimuthal directions and careful data correction, most of this bias can be removed. The results indicate a mostly orogen parallel upper and orogen perpendicular lower layer of anisotropies. In the forelands, we find good agreement with SKS studies from which we infer that lithospheric and asthenospheric anisotropies are mostly parallel. (2) Azimuthally anisotropic 3D shear velocity structure of the eastern Alps from rjMcMC tomography. With this innovative approach, we go beyond what is is shown in (1) and are able resolve the depth structure of the azimuthal anisotropy and estimate the model uncertainties. It is shown that under the orogen, a two layer anisotropic structure exists that separates the upper crust which is dominated by arc-parallel anisotropy from the lower crust and uppermost mantle which mostly show arc-perpendicular fast axis. We find that the anisotropy in the upper crust is largely controlled by major fault structures. The isotropic velocity distribution indicates a fast anomaly in the Giudicarie zone that may be related to Permian magmatism and causes a small offset in the Moho proxy. The estimated Moho structure closely resembles the positions of the underlying subudction slabs with a lateral offset between Central and Eastern Alps. (3) 3D joint inversion of surface and body wave data. To better image the crust-mantle transition zone under the Alps, we apply a 3D rjMcMC imaging approach that combines different datasets and resolves the Vp and Vs structure from the surface to 600 km depth. With this approach there is no need for crustal corrections applied to the body wave travel times since the crustal structure is constrained by ambient noise data. Preliminary results of this model indicate that the slabs are more vertical and vertically more continuous as compared to a pure P-traveltime inversion. The Python scripts used to obtain the results are already or will be published on the author’s github page (github.com/ekaestle)

Lichen Accumulation of Some Heavy Metals from Acidic Surface Substrates of Coal Mine Ecosystems in Southeastern Ohio

Author Institution: Department of Botany, The Ohio State Universit

Fossil rootlet biopores as conduits for contaminant transport through clay horizons: a case study of DNAPL behaviour in Severn alluvium, UK

This paper presents a case study of several DNAPL contaminated sites around the Severn Estuary (UK) where a combination of detailed observations, core dissections and physicochemical characterisation of alluvial clay–silt horizons have revealed the presence of fossil rootlet biopores which act and have the potential to act as conduits for contaminant migration through up to 13 m of clay–silt. The biopores are shown to penetrate the low-permeability (K ~ 10−10 m/s) clay–silt matrix throughout its entire depth (up to 13 m) and provide a preferential transport pathway for DNAPLs from near surface to the underlying aquifer, with particularly high concentrations measured in the biopores themselves. Capillary rise experiments with coal tar distillate demonstrate that DNAPLs are drawn into biopores, with values of surface interfacial tension for the system calculated as 8 × 10−2 J/m2. Wicking by residual plant fibres was demonstrated qualitatively and is thought to be an important additional transport mechanism. The DNAPL contamination below and throughout the Severn alluvial clay–silt horizons demonstrates that the assumption that these horizons act as an effective seal protecting underlying aquifers from severe pollution from the legacy sites around the Severn is flawed and highlights the failure of current protocols for sampling of clay horizons for hydraulic conductivity assessments, since current methods can destroy delicate in situ biopore structures. The study demonstrates that the interplay between ecological and the geological depositional environment can, at a regional scale, result in a network of biopores that can potentially act as conduits for contaminant transport