Remarks . In the case r = 1 the simple lower bound Vj (x) > x(x) is available . When r = 2, the analogous result is V2(x) > 7c 2 (x) where 7r 2 (x) denotes the number of primes p < x such that (p 1)/2 is prime . Evidently the numbers 02(p) = (p -3)/2 are distinct . Sieve theory suggests, but of course does not yet prove, that 7r 2(x) > x/log 2 x, so that apart from the second factor on the righ...

Whittaker 's sampling theorem (Whittaker, 1915; McNamee et al., 1971) for the reconstruction of a frequency-band-limited time-function g(t), f rom its uniform (i.e., equidistant) samples g(k); k ~ J (the class of integers); has been extended by Fogel (1955) and Jagerman and Fogel (1956) to include the simultaneous sampling of g(t) and its first derivative. Fogel's work (1955) is quite incomplet...

CONTINUING THE RECENT SERIES of articles on the pathogenesis of lipopolysaccharide (LPS)-induced fever (8, 9, 11–16, 21, 25, 26), a study by Feleder et al. (7) in this issue of the American Journal of Physiology-Regulatory, Integrative and Comparative Physiology reports a novel finding: an exaggeration of the febrile response to LPS in guinea pigs after splenectomy. The authors attribute this e...

The observed distribution of IMF shapes can be understood as statistical sampling from a universal IMF and variations that result from stellar-dynamical processes. However, young star clusters appear to have an IMF biased towards low-mass stars when compared to the Galactic disk IMF, which comprises an average populated by star-formation episodes occurring Gyrs ago. In addition to this tentativ...

Let s(n) denote the sum of the positive divisors of n except for n iteself. Discussed since Pythagoras, s may be the first function of mathematics. Pythagoras also suggested iterating s, so perhaps considering the first dynamical system. The historical legacy has left us with some colorful and attractive problems, mostly still unsolved. Yet the efforts have been productive in the development of...