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We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$.
The main indicator adopted to track universal salt iodization has been the coverage of adequately iodized salt in households. Rapid test kits (RTK) have been included in household surveys to test the iodine content in salt. However, laboratory studies of their performance have concluded that RTK are reliable only to distinguish between the presence and absence of iodine in salt, but not to determine whether salt is adequately iodized. The aim of the current paper was to examine the performance of RTK under field conditions and to recommend their most appropriate use in household surveys.
Design
Standard performance characteristics of the ability of RTK to detect the iodine content in salt at 0 mg/kg (salt with no iodine), 5 mg/kg (salt with any added iodine) and 15 mg/kg (‘adequately’ iodized salt) were calculated. Our analysis employed the agreement rate (AR) as a preferred metric of RTK performance.
Setting/Subjects
Twenty-five data sets from eighteen population surveys which assessed household iodized salt by both the RTK and a quantitative method (i.e. titration or WYD Checker) were obtained from Asian (nineteen data sets), African (five) and European (one) countries.
Results
In detecting iodine in salt at 0 mg/kg, the RTK had an AR>90 % in eight of twenty-three surveys, while eight surveys had an AR<80 %. When the RTK was used for detecting adequately iodized salt, the AR decreased significantly, with only one of fourteen surveys achieving an AR>90 %.
Conclusions
The RTK is not suited for assessment of adequately iodized salt coverage. Quantitative assessment, such as by titration or WYD Checker, is necessary for estimates of adequately iodized salt coverage.
To compare thyroid-stimulating hormone (TSH) levels in neonatal cord blood between study sites in Bangladesh, Guatemala and the United States. Also, to compare neonatal TSH results with indicators of iodine deficiency in school children.
Design:
Consecutive births and, in school children, cross-sectional surveys.
Setting:
Savar, Bangladesh; San Pedro Sacatepequez, Guatemala; and Atlanta, United States.
Subjects:
In each study site, cord blood was spotted on to filter paper and TSH levels determined using a sensitive monoclonal assay. In the USA, heel stick blood specimens from newborns spotted on to filter paper were also obtained as well as exposure to iodine-containing antiseptics during the birthing process. Urine specimens were collected from mothers of newborns and tested for iodine concentration. School children in the same areas were surveyed for thyroid size by palpation and ultrasonography, and urine specimens collected for iodine concentration.
Results:
Between 141 and 243 cord blood specimens were collected from each study site. The prevalence of elevated cord blood TSH levels (>5 mU l−1) was high in all study sites, from 58% to 84%. All sites would be categorised as having ‘severe’ iodine deficiency based on WHO/UNICEF/ICCIDD criteria. Iodine-containing antiseptics were used during 98% of the births in the USA but not in Bangladesh or Guatemala. The neonatal TSH classification indicated more severe iodine deficiency levels than classifications based on urinary iodine and goitre in school children.
Conclusions:
In the USA, elevated TSH levels may be partially attributed to use of beta-iodine-containing antiseptics prior to birth. We recommend the cautious interpretation of TSH results in newborns for the assessment of iodine deficiency disorders when iodine-containing antiseptics are used during the birthing process.
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