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

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

Optimal Quantum Measurements of Expectation Values of Observables

Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |psi> of the quantum system. Such expectation values can be measured by repeatedly preparing |psi> and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1/sqrt(N) for N repetitions of the experiment. For the problem of estimating the parameter phi in an evolution exp(-i phi H), it is possible to achieve precision 1/N (the quantum metrology limit) provided that sufficient information about H and its spectrum is available. We consider the more general problem of estimating expectations of operators A with minimal prior knowledge of A. We give explicit algorithms that approach precision 1/N given a bound on the eigenvalues of A or on their tail distribution. These algorithms are particularly useful for simulating quantum systems on quantum computers because they enable efficient measurement of observables and correlation functions. Our algorithms are based on a method for efficiently measuring the complex overlap of |psi> and U|psi>, where U is an implementable unitary operator. We explicitly consider the issue of confidence levels in measuring observables and overlaps and show that, as expected, confidence levels can be improved exponentially with linear overhead. We further show that the algorithms given here can typically be parallelized with minimal increase in resource usage.Comment: 22 page

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

New derivation for the equations of motion for particles in electromagnetism

We present equations of motion for charged particles using balanced equations, and without introducing explicitly divergent quantities. This derivation contains as particular cases some well known equations of motion, as the Lorentz-Dirac equations. An study of our main equations in terms of order of the interaction with the external field conduces us to the Landau-Lifshitz equations. We find that the analysis in second order show a special behavior. We give an explicit presentation up to third order of our main equations, and expressions for the calculation of general orders.Comment: 11 pages, 2 figures. Minor changes. Closer to published versio

Floristic composition of Azorean littoral rock pools.

43rd European Marine Biology Symposium”. Ponta Delgada, Açores, 8-12 de Setembro de 2008