Black hole thermodynamical entropy

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy $S_{BG}$ of a $(3+1)$ black hole is proportional to its area $L^2$ ($L$ being a characteristic linear length), and not to its volume $L^3$. Similarly it exists the \emph{area law}, so named because, for a wide class of strongly quantum-entangled $d$-dimensional systems, $S_{BG}$ is proportional to $\ln L$ if $d=1$, and to $L^{d-1}$ if $d>1$, instead of being proportional to $L^d$ ($d \ge 1$). These results violate the extensivity of the thermodynamical entropy of a $d$-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is \emph{not} to be identified with the BG {\it additive} entropy but with appropriately generalized {\it nonadditive} entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.Comment: 7 pages, 2 figures. Accepted for publication in EPJ

Biology and etiology of young-onset breast cancers among premenopausal African American women: Results from the AMBER Consortium

Background: African American (AA) women have higher incidence of aggressive, young-onset (<40 years) breast cancers. Young- and older-onset disease may have distinct tumor biologies and etiologies; however, studies investigating age differences among AA women have been rare and generally underpowered. Methods: We examined tumor characteristics and breast cancer risk factors associated with premenopausal young (<40) vs. older (40) AA women's breast cancer in the African American Breast Cancer Epidemiology and Risk Consortium (2,008 cases and 5,144 controls). Unconditional logistic regression models assessed heterogeneity of tumor biology and risk factor associations by age, overall, and by estrogen receptor status. Results: Premenopausal AA women <40 years had higher frequency of poorer-prognosis tumor characteristics compared with older women, including negative estrogen and progesterone receptor status, triple-negative subtype, higher grade, higher stage, and larger tumors. Adiposity (i.e., waist-to-hip ratio) and family history of breast cancer were more strongly associated with young-onset disease [case–control OR ¼ 1.46, 95% confidence interval (CI) ¼ 1.04–2.05; OR ¼ 3.10, 95% CI ¼ 2.08–4.63, respectively] compared with older-onset disease (OR ¼ 1.11, 95% CI ¼ 0.91–1.35; OR ¼ 1.57, 95% CI ¼ 1.26–1.94). Breastfeeding showed a slight inverse risk association among young women (OR ¼ 0.70, 95% CI ¼ 0.43–1.16). Oral contraceptive use was associated with increased risk regardless of age. Considering various cutoff points for young age (<40, <45, <50), age-related heterogeneity was greatest when <40 was used. Conclusions: Among premenopausal AA women, diagnosis before age 40 is associated with more aggressive breast tumor biology and some etiologic differences. Impact: Modifiable risk factors including breastfeeding, adiposity, and oral contraceptive use may be important targets for mitigating harms of young-onset breast cancer

Tracing the Distribution of European Lactase Persistence Genotypes Along the Americas

In adulthood, the ability to digest lactose, the main sugar present in milk of mammals, is a phenotype (lactase persistence) observed in historically herder populations, mainly Northern Europeans, Eastern Africans, and Middle Eastern nomads. As the –13910∗T allele in the MCM6 gene is the most well-characterized allele responsible for the lactase persistence phenotype, the –13910C &gt; T (rs4988235) polymorphism is commonly evaluated in lactase persistence studies. Lactase non-persistent adults may develop symptoms of lactose intolerance when consuming dairy products. In the Americas, there is no evidence of the consumption of these products until the arrival of Europeans. However, several American countries’ dietary guidelines recommend consuming dairy for adequate human nutrition and health promotion. Considering the extensive use of dairy and the complex ancestry of Pan-American admixed populations, we studied the distribution of –13910C &gt; T lactase persistence genotypes and its flanking haplotypes of European origin in 7,428 individuals from several Pan-American admixed populations. We found that the –13910∗T allele frequency in Pan-American admixed populations is directly correlated with allele frequency of the European sources. Moreover, we did not observe any overrepresentation of European haplotypes in the –13910C &gt; T flanking region, suggesting no selective pressure after admixture in the Americas. Finally, considering the dominant effect of the –13910∗T allele, our results indicate that Pan-American admixed populations are likely to have higher frequency of lactose intolerance, suggesting that general dietary guidelines deserve further evaluation across the continent