Charles F. Osgood

Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better

Abstract This paper will do the following: (1) Establish a (better than) Thue-Siegel-Roth-Schmidt theorem bounding the approximation of solutions of linear differential equations over valued differential fields; (2) establish an effective better than Thue-Siegel-Roth-Schmidt theorem bounding the approximation of irrational algebraic functions (of one variable over a constant field of characteristic zero) by rational functions; (3) extend Nevanlinnas Three Small Function Theorem to an n small function theorem (for each positve integer n), by removing Chuangs dependence of the bound upon the relative “number” of poles and zeros of an auxiliary function; (4) extend this n Small Function Theorem to the case in which the n small functions are algebroid (a case which has applications in functional equations); (5) solidly connect Thue-Siegel-Roth-Schmidt approximation theory for functions with many of the Nevanlinna theories. The method of proof is (ultimately) based upon using a Thue-Siegel-Roth-Schmidt type auxiliary polynomial to construct an auxiliary differential polynomial.

An effective lower bound on the “Diophantine approximation” of algebraic functions by rational functions

This chapter reviews an effective lower bound on the diophantine approximation of algebraic functions by rational functions. In an article in the Proceedings in 1959, Professor Ellis Kolchin showed an analogue of Liouvilles theorem, which dealt with the approximation of solutions of algebraic differential equations. The most concrete application of Kolchins theorem is to the approximation by rational functions of formal power series solutions of a nontrivial algebraic differential equation. Kolchins theorem says that there exists a natural number n and a real number c such that for every r , s ≠ 0: ord( y 1 - r/s ) n (deg s ) + c . Here, y 1 is not a rational function of z but is a zero of a not identically zero polynomial α. An integer n , which will suffice in the above theorem, may be read off from any nonidentical zerodifferential polynomial α of which y l is a nonsingular zero. Thus, an n for the above theorem may be effectively computed.

On the quotient of two integral functions

Remark. Rubel suggested the more general conjecture that any function algebraic over H, if entire, belongs to H. The proof of Theorem 1 has something to do with the distribution of the zeros of an exponential polynomial and the convex hull of the 0~~‘s. Thus, the methods used in [4] by their nature will fail to deal with the situation that some or all of the q’s are replaced by polynomials. In this paper, we shall treat a special case of the above-mentioned situation, As an application of the study, we shall present a result concerning the zero-one set problem (see [5]) for the class of all integral functions of finite order. Let us denote by G the class of all integral functions of the form:

On the equation P(x) = n! and a question of Erdős

Abstract Given a polynomial P of degree ≥ 2 with integer coefficients, it is shown that the set of n , for which the equation P ( x ) = n ! has an integer solution x , is of density zero. This answers in particular a question of Erdős.

Nearly perfect systems and effective generalizations of Shidlovski's Theorem

Abstract An effective proof of Shidlovskis Theorem is presented. The proof utilizes partial differential operators. A number of generalizations of Shidlovskis Theorem are proven, including results about approximation at more than one point. Additionally, partial differential equations are considered. The new methods give a particularly direct proof of Shidlovskis Theorem.

Infinite Riemann Sums, the Simple Integral, and the Dominated Integral

Simple integrability of a function f (defined by Haber and Shisha in [2]) is shown to be equivalent to the convergence of the infinite Riemann sum

On the entire solutions of a functional equation in the theory of fluids

\sum\limits_{k = 1}^\infty {f\left( {{\xi _k}} \right)\left( {{x_k} - {x_{k - 1}}} \right)}

Primeable entire functions

to the improper Riemann integral \( \int_0^\infty f \) f as the gauge of the partition \( \left( {{x_k}} \right)_{k = 0}^\infty \) of [0,∞)converges to O. An analogous result is obtained for dominant integrability (defined by Osgood and Shisha in [5]). Also certain results of Bromwich and Hardy [1] are recovered.