Textbook accounts of the rules of indices with rational exponents

The rules of indices, e.g. anbn = (ab)n, are a particularly important part of elementary algebra. This paper reports results from a textbook analysis which examined how the shift from integer to rational exponents in the rules of indices is discussed in school textbooks. The analysis also considered related issues, such as notation and the introduction of complex numbers. A selection of popular textbooks from the period 1800–2000 was examined and the nature of the justification given for the extension of meaning to rational indices considered. In both the definition and computational rules, when extending the domain of n in an to rational numbers the (potential) contraction of the domain of a to positive numbers was often quietly ignored. A wide variety of approaches are used in choosing what is to be a definition, what is to follow, and how this is justified. The difference between computational rules for practical algebraic manipulation and a formal definition was often blurred.<br/

Delayed Treatment of Diagnosed Pulmonary Tuberculosis in Taiwan

<p>Abstract</p> <p>Background</p> <p>Mycobacterium tuberculosis infection is an ongoing public health problem in Taiwan. The National Tuberculosis Registry Campaign, a case management system, was implemented in 1997. This study examined this monitoring system to identify and characterize delayed treatment of TB patients.</p> <p>Methods</p> <p>Records of all tuberculosis cases treated in Taiwan from 2002 through 2005 were obtained from the National Tuberculosis Registry Campaign. Initiation of treatment more than 7 days after diagnosis was considered a long treatment delay.</p> <p>Results</p> <p>The study included 31,937 patients. The mean day of delayed treatment was 3.6 days. Most patients were treated immediately after diagnosis. The relationship between number of TB patients and days of delayed treatment after diagnosis exhibited a Power-law distribution. The long tail of the power-law distribution indicated that an extreme number occur cannot be neglected. Tuberculosis patients treated after an unusually long delay require close observation and follow up.</p> <p>Conclusion</p> <p>This study found that TB control is generally acceptabl in Taiwan; however, delayed treatment increases the risk of transmission. Improving the protocol for managing confirmed TB cases can minimize disease transmission.</p

Genome-wide association analysis identifies novel blood pressure loci and offers biological insights into cardiovascular risk

Elevated blood pressure is the leading heritable risk factor for cardiovascular disease worldwide. We report genetic association of blood pressure (systolic, diastolic, pulse pressure) among UK Biobank participants of European ancestry with independent replication in other cohorts, and robust validation of 107 independent loci. We also identify new independent variants at 11 previously reported blood pressure loci. In combination with results from a range of in silico functional analyses and wet bench experiments, our findings highlight new biological pathways for blood pressure regulation enriched for genes expressed in vascular tissues and identify potential therapeutic targets for hypertension. Results from genetic risk score models raise the possibility of a precision medicine approach through early lifestyle intervention to offset the impact of blood pressure-raising genetic variants on future cardiovascular disease risk

The genetic architecture of type 2 diabetes.

The genetic architecture of common traits, including the number, frequency, and effect sizes of inherited variants that contribute to individual risk, has been long debated. Genome-wide association studies have identified scores of common variants associated with type 2 diabetes, but in aggregate, these explain only a fraction of the heritability of this disease. Here, to test the hypothesis that lower-frequency variants explain much of the remainder, the GoT2D and T2D-GENES consortia performed whole-genome sequencing in 2,657 European individuals with and without diabetes, and exome sequencing in 12,940 individuals from five ancestry groups. To increase statistical power, we expanded the sample size via genotyping and imputation in a further 111,548 subjects. Variants associated with type 2 diabetes after sequencing were overwhelmingly common and most fell within regions previously identified by genome-wide association studies. Comprehensive enumeration of sequence variation is necessary to identify functional alleles that provide important clues to disease pathophysiology, but large-scale sequencing does not support the idea that lower-frequency variants have a major role in predisposition to type 2 diabetes

Rare and low-frequency coding variants alter human adult height

Height is a highly heritable, classic polygenic trait with approximately 700 common associated variants identified through genome-wide association studies so far. Here, we report 83 height-associated coding variants with lower minor-allele frequencies (in the range of 0.1-4.8%) and effects of up to 2 centimetres per allele (such as those in IHH, STC2, AR and CRISPLD2), greater than ten times the average effect of common variants. In functional follow-up studies, rare height-increasing alleles of STC2 (giving an increase of 1-2 centimetres per allele) compromised proteolytic inhibition of PAPP-A and increased cleavage of IGFBP-4 in vitro, resulting in higher bioavailability of insulin-like growth factors. These 83 height-associated variants overlap genes that are mutated in monogenic growth disorders and highlight new biological candidates (such as ADAMTS3, IL11RA and NOX4) and pathways (such as proteoglycan and glycosaminoglycan synthesis) involved in growth. Our results demonstrate that sufficiently large sample sizes can uncover rare and low-frequency variants of moderate-to-large effect associated with polygenic human phenotypes, and that these variants implicate relevant genes and pathways.</p