Thin Sequences and Their Role in HpH^p Theory, Model Spaces, and Uniform Algebras

In this paper we revisit some facts about thin interpolating sequences in the unit disc from three perspectives: uniform algebras, model spaces, and $H^p$ spaces. We extend the notion of asymptotic interpolation to $H^p$ spaces, for $1 \leq p \leq \infty$, providing several new ways to think about these sequences.Comment: v1: 21 pages; To appear in Rev. Mat. Iberoa

Pervasive Algebras and Maximal Subalgebras

A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of the lattice of all subsets of $\N$ into the family of pervasive subalgebras of some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is removed. We discuss further examples, involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures

Microfluidic system with a wireless paired emitter detector diode device as optical sensor for water quality monitoring

Increased demand for improved water management is driving need for water quality monitoring systems with greatly improved price/performance characteristics. This work presents the first use of wireless paired emitter detector diode device (PEDD) as an optical sensor for colorimetric analysis of water quality in a Lab-on-a-disc device format. The instrument detector involves using two light emitting diodes (LEDs), which act as both a light source and photo detector (Fig. 1a.). In comparison to the more commonly used method of coupling a LED to a photodiode, this technique achieves excellent sensitivity and signal-to-noise ratio, with very low cost fabrication and electronics. Furthermore, its low power consumption, increasing spectral range coverage, excellent intensity and efficiency, small size, ease of fabrication and simplicity of the PEDD make it a perfect optical detector for colorimetric assays [1]. In addition, the device is ideally suited for integration with microfluidic platforms based on the centrifugal Lab-on-a-Disc concept, in which detector difficulties can arise due to the high rotation speed typically used in this approach [2]. In this work the calibration of the system using bromocresol purple (BCP) is demonstrated. Concentration ranges were examined in parallel using UV-Vis spectroscopy as control, and the PEDD system. Similar limits of detection (ca. 2.5x10-4 M, Fig.1b.) were obtained in both cases. However, the PEDD system presented a linear trend over a wider range of concentrations. The experiments demonstrate the potential for the wireless PEDD to be a versatile and cheap alternative optical detector system for water quality monitoring in microfluidic applications