Distribution of α p 2 Modulo One with Prime Variable p of a Special Form

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for α ∈ ℝ / ℚ , β ∈ ℝ , and 0 < θ < 10 / 1561 , there exist infinitely many primes p , such that α p 2 + β < p − θ and p + 2 = P 4 , which constitutes an improvement upon the previous result.

A Unified Correlation for Slip Factor in Centrifugal Impellers

A method that unifies the trusted centrifugal impeller slip factor prediction methods of Busemann, Stodola, Stanitz, Wiesner, Eck, and Csanady in one equation is presented. The simple analytical method derives the slip velocity in terms of a single relative eddy (SRE) centered on the rotor axis instead of the usual multiple (one per blade passage) eddies. It proposes blade solidity (blade length divided by spacing at rotor exit) as the prime variable determining slip. Comparisons with the analytical solution of Busemann and with tried and trusted methods and measured data show that the SRE method is a feasible replacement for the well-known Wiesner prediction method: it is not a mere curve fit, but is based on a fluid dynamic model; it is inherently sensitive to impeller inner-to-outer radius ratio and does not need a separate calculation to find a critical radius ratio; and it contains a constant, F0, that may be adjusted for specifically constructed families of impellers to improve the accuracy of the prediction. Since many of the other factors that contribute to slip are also dependent on solidity, it is recommended that radial turbomachinery investigators and designers investigate the use of solidity to correlate slip factor.

On the distribution of αpk modulo 1

The fractional part of the sequence {αnk}, where α is an irrational real number and k is an integer, was first studied early this century, initiated by the work of Hardy, Littlewood and Weyl. It seems very natural to consider the subsequence {αpk}, where p denotes a prime variable. The pioneering work in this direction was conducted by Vinogradov [13,14]. Improvements have since been made by Vaughan [12], Ghosh [4], Harman [6,7,8] and Jia [11]. The best results to date have been obtained by Harman for k = 1 [9], by Baker and Harman for 2 ≤ k ≤ 12 [1], and by Harman for larger k [8]. In the following work, we shall adopt a sieve technique developed by Harman in [6] to show the following.

Numbers badly approximate by fractions with prime denominator

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).