Ramanujan and Extensions and Contractions of Continued Fractions

If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of $K_{n=1}^{\infty} a_{n}/b_{n}$ we mean a continued fraction $K_{n=1}^{\infty} c_{n}/d_{n}$ whose odd or even part is $K_{n=1}^{\infty} a_{n}/b_{n}$. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if $K_{n=1}^{\infty}a_{n}/b_{n}$ is the even part); (ii) Find the limit of the other contraction and show that the odd and even parts of the extension tend to the same limit. We apply these ideas to derive new proofs of certain continued fraction identities of Ramanujan and to prove a generalization of an identity involving the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith and Brillhart.Comment: 16 page

Lessons Learned in Pooling Data for Reference Populations

This study set out to demonstrate the feasibility of merging data from 4 different experimental resource dairy populations (1 herd in each of Scotland and Ireland, and 2 in the Netherlands) to create a pooled reference population for joint genetic and genomic analyses. Data included a total of 60,058 weekly records from 1,630 Holstein-Friesian cows across the 4 herds and included 7 traits: milk, fat and protein yield, milk somatic cell count, live weight, dry matter intake, and energy intake and balance. Missing records were predicted using random regression models, so that at the end there were 44 weekly records, corresponding to the typical 305-day lactation, for each cow. Data were subsequently merged and analysed with mixed linear models. Genetic variance and heritability estimates were greater (P0.05) from zero. When estimable, the genetic correlation between herds for the same trait ranged from 0.85 to 0.99. Results suggested that merging experimental herd data into a single dataset is both feasible and sensible, despite potential differences in management and recording of the animals in the four herds. Merging experimental data will increase the precision of parameter estimates in a genetic analysis and augment the potential reference population in genome-wide association studies especially of difficult-to-record traits

Cardiac magnetic resonance imaging analysis in STEMI: quantitative or still visual?

Cardiovascular Aspects of Radiolog

Left ventricular mass assessment by CMR; how to define the optimal index

Vascular Biology and Interventio

Increased accuracy in computed tomography coronary angiography; a new body surface area adapted protocol

Vascular Biology and Interventio

New insights into measurement variability in glaucomatous visual fields from computer modelling.

To develop a model to simulate visual fields (VFs) in glaucoma patients, and to characterize variability of the Mean Deviation (MD) VF summary measurement using real VFs and simulations