An important component of Apéry’s proof that ζ(3) is irrational involves representing as the limit quotient two rational solutions to a three-term recurrence. We present various approaches such Apéry limits and highlight connections continued fractions well famous theorems Poincaré Perron on difference equations. In spirit Jon Borwein, we advertise an experimental mathematics approach by first ...

The van der Waerden number W (k, 2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W (k, 2) are enormous. Much effort was put into developing lower bounds on W (k, 2). Most of these lower bound proofs employ the probabilistic method oft...

In order to compare the quality of proofs, it is necessary to measure artifacts of the proofs, and evaluate the measurements to determine differences between the proofs. This paper discounts the approach of ranking proofs by their measured proof artifacts, and takes the position that different proofs are good proofs. The position is based on proofs in the TSTP solution library, which are genera...

Bitcoin’s innovative and distributedly maintained blockchain data structure hinges on the adequate degree of difficulty of so-called “proofs of work,” which miners have to produce in order for transactions to be inserted. Importantly, these proofs of work have to be hard enough so that miners have an opportunity to unify their views in the presence of an adversary who interferes but has bounded...

Different types of mathematical inequalities have been largely analyzed and employed. In this paper, we introduce improvements to some Ostrowski type present their corresponding proofs. The presented proofs are based on applying the celebrated Hayashi inequality certain functions. We provide examples that show these improvements. Illustrations obtained results stated in a probability framework.

BACKGROUND T1 mapping is widely used today in CMR, however, it underestimates true T1 values and its measurement error is influenced by several acquisition parameters. The purpose of this study was the extraction of accurate T1 data through the utilization of comprehensive, parallel Simulations for QUAntifying RElaxation Magnetic Resonance constants (SQUAREMR) of the MOLLI pulse sequence on a l...

Flocks participating in sire referencing schemes can achieve greater genetic gains than those achievable by within-flock selection. However, requirements for joining these schemes can be prohibitive to some producers. The objectives of this study were to determine whether less restrictive schemes or schemes of shorter duration could achieve rates of gain and reduce inbreeding as efficiently as ...

While holding great development opportunities to be seized, Africa has been identified as particularly vulnerable to climate changes, and the ZRB is no exception to that. With increasing demands to the Zambezi system in terms or water and energy, and rising challenges in the form of less runoff due to climate change, it is paramount to plan ahead. In particular, it must be understood how major ...