The Chernoff-Hoeffding bounds are fundamental probabilistic tools. An elementary approach is presented to obtain a Chernoff-type upper-tail bound for the number of prime factors of a random integer in {1, 2, . . . , n}. The method illustrates tail bounds in negatively-correlated settings.

We discuss five ways of proving Chernoff’s bound and show how they lead to different extensions of the basic bound.

Alpha-Hydroxy ketones with an aromatic and aliphatic substituent were prepared with condensation of aromatic and aliphatic aldehydes in the presence of polymer-bound thiazolium salt.

We construct a lot of new [n, 4, d]9 codes whose lengths are close to the Griesmer bound and prove the nonexistence of some linear codes attaining the Griesmer bound using some geometric techniques through projective geometries to determine the exact value of n9(4, d) or to improve the known bound on n9(4, d) for given values of d, where nq(k, d) is the minimum length n for which an [n, k, d]q ...

We establish local (L, L) mapping properties for averages on curves. The exponents are sharp except for endpoints.

Using a new method we construct a class of orthogonal designs, o D( 4(1 + P )n; n, n, n, n, pn, pn, pn, pn) for p = 1, p a prime power == 3 (mod 4), and n = 3,5,7. This class includes new Plotkin arrays of order 24, and for the first time, of orders 40 and 56.