On the fractal characteristics of a stabilised Newton method

In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed

Breastfeeding, lactational infecundity, contraception and the spacing of births: implications of the Bellagio Consensus Statement

While the nutritional, immunological and anti-bacterial benefits of breast milk are incontestable, the contraceptive effect of breastfeeding is more apparent in the aggregate than at the level of the individual. Nevertheless, it has recently been recommended that lactating women not consider adopting contraception until the earliest of their first post-partum menstruation, the introduction of supplementary feeding or their child’s reaching six months of age. This article employs microsimulation to quantify the implications of this recommendation for the spacing of births and, in particular, for the proportion of birth intervals that are unacceptably short. The findings are not encouraging. The implementation of this protocol would not produce better birth spacing than a simpler strategy of initiating contraception early in the post-partum period and, unless implemented perfectly, the outcomes would be considerably worse. Breastfeeding should be viewed not as a method of birth control but as the best form of infant nourishment. Efficient contraception is the best way to ensure that children in modernizing societies can reap the benefits of breastfeeding, without being endangered by being weaned too early because of a new pregnancy

Fractal Characteristics of Newton's Method on Polynomials

In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to find the roots of a general complex polynomial with real coefficients. For the case of symmetric polynomials $z^{\nu}-1$ , the generation mechanism is derived from first principles. We discuss the case of a general cubic and are able to give a description of the arising fractal structure depending on the coefficients of the cubic. Special cases are analysed and their characteristics, including scale factors and an approximate fractal dimension, are derived. The theoretical results are confirmed via computational experiments. An application of the theory in turbulence modelling is presented

Exploring experiences of cancer care in Wales: a thematic analysis of free-text responses to the 2013 Wales Cancer Patient Experience Survey (WCPES)

Objectives To provide the first systematic analysis of a national (Wales) sample of free-text comments from patients with cancer, to determine emerging themes and insights regarding experiences of cancer care in Wales. Design Thematic analysis of free-text data from a population-based survey. Setting and participants Adult patients with a confirmed cancer diagnosis treated within a 3-month period during 2012 in the 7 health boards and 1 trust providing cancer care in Wales. Main outcome measures Free-text categorised by theme, coded as positive or negative, with ratios. Overarching themes are identified incorporating comment categories. Methods 4672 respondents (of n=7352 survey respondents) provided free-text comments. Data were coded using a multistage approach: (1) coding of comments into general categories (eg, nursing, surgery, etc), (2) coding of subcategories within main categories (eg, nursing care, nursing communication, etc), (3) cross-sectional analysis to identify themes cutting across categories, (4) mapping of categories/subcategories to corresponding closed questions in the Wales Cancer Patient Experience Survey (WCPES) data for comparison. Results Most free-text respondents (82%, n 3818) provided positive comments about their cancer care, with 49% (n=2313) giving a negative comment (ratio 0.6:1, negative-to-positive). 3172 respondents (67.9% of free-text respondents) provided a comment mapping to 1 of 4 overarching themes: communication (n=1673, 35.8% free-text respondents, a ratio of 1.0:1); waiting during the treatment and/or post-treatment phase (n=923, 19.8%, ratio 1.5:1); staffing and resource levels (n=671, 14.4% ratio 5.3:1); speed and quality of diagnostic care (n=374, 8.0%, ratio 1.5:1). Within these areas, constituent subthemes are discussed. Conclusions This study presents specific areas of concern for patients with cancer, and reveals a number of themes present across the cancer journey. While the majority of comments were positive, analysis reveals concerns shared by significant numbers of respondents. Timely communication can help to manage these anxieties, even where delays or difficulties in treatment may be encountered