On a class of relatively prime sequences

AbstractFor each natural number n, let a0(n) = n, and if a0(n),…,ai(n) have already been defined, let ai+1(n) > ai(n) be minimal with (ai+1(n), a0(n) … ai(n)) = 1. Let g(n) be the largest ai(n) not a prime or the square of a prime. We show that g(n) ∼ n and that g(n) > n + cn12log(n) for some c > 0. The true order of magnitude of g(n) − n seems to be connected with the fine distribution of prime numbers. We also show that “most” ai(n) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our proofs: For every sufficiently large n there is a prime p < n12 with [np] composite

An infinite family of convex Brunnian links in RnR^n

This paper proves that convex Brunnian links exist for every dimension $n \geq 3$ by constructing explicit examples. These examples are three-component links which are higher-dimensional generalizations of the Borromean rings.Comment: 10 pages, 4 figure

Partially Annealed Disorder and Collapse of Like-Charged Macroions

Charged systems with partially annealed charge disorder are investigated using field-theoretic and replica methods. Charge disorder is assumed to be confined to macroion surfaces surrounded by a cloud of mobile neutralizing counterions in an aqueous solvent. A general formalism is developed by assuming that the disorder is partially annealed (with purely annealed and purely quenched disorder included as special cases), i.e., we assume in general that the disorder undergoes a slow dynamics relative to fast-relaxing counterions making it possible thus to study the stationary-state properties of the system using methods similar to those available in equilibrium statistical mechanics. By focusing on the specific case of two planar surfaces of equal mean surface charge and disorder variance, it is shown that partial annealing of the quenched disorder leads to renormalization of the mean surface charge density and thus a reduction of the inter-plate repulsion on the mean-field or weak-coupling level. In the strong-coupling limit, charge disorder induces a long-range attraction resulting in a continuous disorder-driven collapse transition for the two surfaces as the disorder variance exceeds a threshold value. Disorder annealing further enhances the attraction and, in the limit of low screening, leads to a global attractive instability in the system.Comment: 21 pages, 2 figure

Observational and genetic associations between cardiorespiratory fitness and cancer: a UK Biobank and international consortia study

Background The association of fitness with cancer risk is not clear. Methods We used Cox proportional hazards models to estimate hazard ratios (HRs) and 95% confidence intervals (CIs) for risk of lung, colorectal, endometrial, breast, and prostate cancer in a subset of UK Biobank participants who completed a submaximal fitness test in 2009-12 (N = 72,572). We also investigated relationships using two-sample Mendelian randomisation (MR), odds ratios (ORs) were estimated using the inverse-variance weighted method. Results After a median of 11 years of follow-up, 4290 cancers of interest were diagnosed. A 3.5 ml O2⋅min−1⋅kg−1 total-body mass increase in fitness (equivalent to 1 metabolic equivalent of task (MET), approximately 0.5 standard deviation (SD)) was associated with lower risks of endometrial (HR = 0.81, 95% CI: 0.73–0.89), colorectal (0.94, 0.90–0.99), and breast cancer (0.96, 0.92–0.99). In MR analyses, a 0.5 SD increase in genetically predicted O2⋅min−1⋅kg−1 fat-free mass was associated with a lower risk of breast cancer (OR = 0.92, 95% CI: 0.86–0.98). After adjusting for adiposity, both the observational and genetic associations were attenuated. Discussion Higher fitness levels may reduce risks of endometrial, colorectal, and breast cancer, though relationships with adiposity are complex and may mediate these relationships. Increasing fitness, including via changes in body composition, may be an effective strategy for cancer prevention

An integrative multi-omics analysis to identify candidate DNA methylation biomarkers related to prostate cancer risk

It remains elusive whether some of the associations identified in genome-wide association studies of prostate cancer (PrCa) may be due to regulatory effects of genetic variants on CpG sites, which may further influence expression of PrCa target genes. To search for Cp

Identification of multiple risk loci and regulatory mechanisms influencing susceptibility to multiple myeloma

Genome-wide association studies (GWAS) have transformed our understanding of susceptibility to multiple myeloma (MM), but much of the heritability remains unexplained. We report a new GWAS, a meta-analysis with previous GWAS and a replication series, totalling 9974 MM cases and 247,556 controls of European ancestry. Collectively, these data provide evidence for six new MM risk loci, bringing the total number to 23. Integration of information from gene expression, epigenetic profiling and in situ Hi-C data for the 23 risk loci implicate disruption of developmental transcriptional regulators as a basis of MM susceptibility, compatible with altered B-cell differentiation as a key mechanism. Dysregulation of autophagy/apoptosis and cell cycle signalling feature as recurrently perturbed pathways. Our findings provide further insight

Repeat Instability in the 27-39 CAG Range of the HD Gene in the Venezuelan Kindreds: Counseling Implications

The instability of the CAG repeat size of the HD gene when transmitted intergenerationally has critical implications for genetic counseling practices. In particular, CAG repeats between 27 and 35 have been the subject of debate based on small samples. To address this issue, we analyzed allelic instability in the Venezuelan HD kindreds, the largest and most informative families ascertained for HD. We identified 647 transmissions. Our results indicate that repeats in the 27-35 CAG range are highly stable. Out of 69 transmitted alleles in this range, none expand into any penetrant ranges. Contrastingly, 14% of alleles transmitted from the incompletely penetrant range (36-39 CAGs) expand into the completely penetrant range, characterized by alleles with 40 or more CAG repeats. At least 12 of the 534 transmissions from the completely penetrant range contract into the incompletely penetrant range of 36-39 CAG repeats. In these kindreds, none of the individuals with 27-39 CAGs were symptomatic, even though they ranged in age from 11 to 82 years. We expect these findings to be helpful in updating genetic counseling practices. © 2008 Wiley-Liss, Inc.link_to_subscribed_fulltex