The Maximal Entropy Measure of Fatou Boundaries

We look at the maximal entropy (MME) measure of the boundaries of connected components of the Fatou set of a rational map of degree greater than or equal to 2. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness

Exactness and maximal automorphic factors of unimodal interval maps

We study exactness and maximal automorphic factors of C^3 unimodal maps of the interval. We show that for a large class of infinitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic non-singular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact examples of this type. We also prove that every C^3 S-unimodal map with no attractor is exact with respect to Lebesgue measure. Additional results about measurable attractors in locally compact metric spaces are given

AMINO ACIDS AS DIAGNOSTICS OF SOIL AND SOIL WATER QUALITY

Information on the contribution of amino acids to dissolved organic nitrogen and carbon exported from grassland soil is scarce. Evidence from the literature for other environments, suggests that determination of amino acid patterns of distribution may be a useful method for improved understanding of the interaction of microbial synthesis and degradation of organic N in conjunction with soil physical states. A sample pre-concentration technique and an HPLC methodology were developed that enabled the determination of dissolved free (DFAA) and combined (DCAA) amino acids in natural waters at picomolar concentration. These methods were used to examine the content of amino acids and their distribution patterns in waters from 3 different settings. Firstly, field-sized lysimeters (1 ha) were used to examine dissolved free and combined amino acids in surface runoff and drainage waters from a grassland soil over 3 winter drainage periods. The waters were collected from soils beneath drained and undrained permanent ryegrass swards, receiving 280 kg N haˉ¹yrˉ¹ , permanent ryegrass receiving no mineral N input, and grass/white clover (no mineral N). Total DFAA concentration ranged between 1.9 nM - 6.1 µM and total DCAA concentration ranged between 1.3 - 87 µM. A large library of amino acid distributions was assembled and multivariate pattern analysis techniques were used to determine whether there were distinctive amino acid signatures that could be used as a diagnostics for soil management and condition. Although addition of mineral N fertilizer increased amino acid concentration in waters, there was no detectable effect of fertilizer addition on DFAA distribution patterns. In contrast, both DFAA and DCAA patterns were strongly influenced by soil hydrology alone. However, in the case of DCAA patterns, there was evidence of an interaction between hydrology and fertilizer addition. Secondly, monolith lysimeters were used to determine the DFAA in drainage waters from 4 different grassland soil types, in order to find whether there was evidence of pattern difference with soil texture. Results showed that distribution patterns vary between soil types, and contrary to what might be expected, that clay soils do not necessarily retain basic amino acids. Thirdly, the concentration and patterns of DFAA were determined hourly over a 24 hour period, for a river that received exported soil waters from the field lysimeters mentioned above. Total DFAA concentration correlated with water temperature and NH4+ Compared with exported soil waters, the concentrations of DFAA in river water were several orders of magnitude smaller, although GLY, SER, LYS and MET were in greater relative proportions. Results of the studies show that amino acids have the potential to be used as diagnostics of source, soil condition and management.BBSRC Institute of Grassland and Environmental Researc

Connectivity properties of Julia sets of Weierstrass elliptic functions

We discuss the connectivity properties of Julia sets of Weierstrass elliptic ℘ functions, accompanied by examples. We give sufficient conditions under which the Julia set is connected and show that triangular lattices satisfy this condition. We also give conditions under which the Fatou set of ℘ contains a toral band and provide an example of an order two elliptic function on a square lattice whose Julia set is a Cantor set

Science-Religion Questions in Science Classrooms

Natural Scienc

A comparative study of water perfusion catheters and microtip transducer catheters for urethral pressure measurements

The aim of this study was to compare the maximum urethral closure pressure (MUCP) measures with two different techniques: water perfused catheter and microtip transducer catheters with respect to reproducibility and comparability for urethral pressure measurements. Eighteen women with stress urinary incontinence had repeat static urethral pressure profilometry on a different day using a dual microtip transducer and water perfused catheter (Brown and Wickham). The investigators were blinded to the results of the other. The microtip measurements were taken in the 45° upright sitting position with the patient at rest at a bladder capacity of 250ml using an 8 Fr Gaeltec® double microtip transducer withdrawn at 1mm/s, and the transducer was orientated in the three o'clock position. Three different measures were taken for each patient. Three water perfusion measurements were performed with the patient at rest in the 45° upright position at a bladder capacity of 250ml using an 8 Fr BARD dual lumen catheter withdrawn at 1mm/s. The mean water perfusion MUCP measure was 26.1cm H20, significantly lower than the mean microtip measure of 35.7cm H20. The correlation coefficient comparing each water perfusion measurement with the other water perfusion measures in the same patient was excellent, at 0.95 (p = 0.01). Correlation coefficient comparing each microtip measure with the other microtip measure in the same patient was also good, ranging from 0.70 to 0.80. This study confirms that both water perfusion catheters and microtip transducers have excellent or very good reproducibility with an acceptable intraindividual variation for both method

A Construction of a Non-Measure-Preserving Endomorphism Using Quotient Relations and Automorphic Factors

The main purpose of this paper is to construct ergodic, nonsingular, conservative n-to-one endomorphisms which preserve no equivalent sfinite measure. While such examples are known to exist w6, 9, 12, 19x, our examples are fundamentally different from the existing ones. It was noticed independently in w6, 19x that the two-sided type III Bernoulli shift of Hamachi w9x is the natural extension of an exact two-to-one endomorphism with no equivalent s-finite invariant measure. Also in w12x it is shown that Cartesian products of finite measure-preserving exact with type III automorphisms yield examples of n-to-one endomorphisms with no equivalent invariant measure. In Section 5 we give an example of a two-to-one endomorphism whose ergodic nonsingular measure is neither exact nor is it equivalent to a product measure of an automorphism with an exact endomorphism