The recurrent construction of combinatorial objects proceeds in steps at each of which we modify the partially constructed object by certain elementary operations such as adjunction of a new element,â€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

1974

Starting from the derivative of the functional equation of the zeta-function, it is shown that at least one-third of the zeros of the zeta-function are on the line sigma = (1/2).

where dx is a a-finite measure on the measure space Q; the real kernel K is given on J.2 x 52; f(r, r) is a given real-valued function on Q x R; h is a given function on Q; and we ask for a solutionâ€¦ (More)

1. The effect of a perturbation on the solutions of a linear system of differential equations can be studied by means of the variation of constants formula. The nonlinear variation of constantsâ€¦ (More)

Proceedings of the National Academy of Sciencesâ€¦

1975

Let s = sigma + it. For any complex a, all but O(1/log log T) of the roots of zeta(s) = a in T < t < 2T lie in|sigma - 1/2| < (log log T)(2)/log T. The results extend easily to other functionsâ€¦ (More)

In this thesis, we present bijections proving partitions identities. In the first part, we generalize Dyson's definition of rank to partitions with successive Durfee squares. We then present twoâ€¦ (More)

It is the purpose of this paper to prove Theorems 1 and 2 below which relate the existence, uniqueness, and general behavior of the solution, y(x, e), for small e>0, of the two-point boundary valueâ€¦ (More)

This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics â€˜number of fixedâ€¦ (More)