The stability problem for functional equations or (partial) differential equations started with the question of Ulam [1]: Under what conditions does there exist an

additive function near an approximately

additive function?

The compounds of interest are shown in Figure 1, with their respective structure, pKa value (logarithmic function of disassociation constant) and

additive function.

Since f(2x) = 2f(x) and f(3x) = [E.sub.4]f(x,-x) + 3f(x) = 3f(x), f is an

additive function.

An attempt to treat the environmental quality as an

additive function of homogenous elements fails to recognize the complexity associated with natural systems and ecological processes.

To Prove the uniqueness of the

additive function A, assume that there exists an

additive function A' : X [right arrow] Y which satisfies (0.4).

In elementary number theory, we call an arithmetical function f (n) as an

additive function, if for any positive integers m, n with (m, n) = 1, we have f (mn) = f (m) + f (n).

Let [Florin] (n) be an arithmetical function, we call [Florin] (n) as an

additive function, if for any positive integers m, n with (m, n) = 1, we have [Florin] (mn) = [Florin] (m) + [Florin] (n).

That is, [bar.[OMEGA]] (n) is the

additive function. The main purpose of this paper is using the elementary methods to study the mean value properties of [bar.[OMEGA]] (n)SL(n), and give a sharper asymptotic formula for it.

Looking in turn at elementary methods, complex analysis methods, and probabilistic methods, he considers such topics as prime numbers, arithmetic functions, sieve methods, the method of van der Corput, the Euler gamma function, summation formulae, the prime number theorem and the Riemann hypothesis, two arithmetic application, primes in arithmetic progressions, densities, distributions of

additive functions and mean values of multiplicative functions, and integers free of small prime factors.

Table 1--Relationship between Dispersion Process Steps and Surface Active

Additive Functions A) surface B) Dynamic C) Robust, tension mobile static reduction stabilization stabilization 1) Wetting of Critical Helpful the dry solid 2) Milling to Helpful Critical Helpful reduce particle size 3) Helpful Critical Stabilization of trie dispersed particle 4) Letdown and Critical application performance D) E) Foam Compatibiization control 1) Wetting of the dry solid 2) Milling to Critical reduce particle size 3) Helpful Stabilization of trie dispersed particle 4) Letdown and Critical application performance Additives used to enhance performance in one step will often have little to no impact on other steps.

Ivic, Estimates for sums involving the largest prime factor of an integer and certain related

additive functions, Studia Sci.

Most of the major

additive functions are covered with the notable exception of gums and thickeners which did receive separate attention in the first edition.