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In this paper we consider perturbations of symmetric Boolean functions $${{\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}}}$$ in n-variable and degree k. We compute the asymptotic behavior of Boolean functions of the type for j fixed. In particular, we characterize all the Boolean functions of the type that are asymptotic balanced. We also present an algorithm that computes the asymptotic behavior of a family of Boolean functions from one member of the family. Finally, as a byproduct of our results, we provide a relation between the parity of families of sums of binomial coefficients. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the properties of Gauss sums and the estimate for character sums to study a mean value problem related to the primitive roots modp and the different forms of Golomb's conjectures and propose an interesting asymptotic formula for it. [ABSTRACT FROM AUTHOR]

In many studies there is believed to be a trend in a categorical variate with a continuous variate. Often in these cases it is desired to test for the trend using k groups farmed from the continuous variable. This article is concerned with the grouping problem where the goal is essentially to maximize the asymptotic efficiency of the test. Optimal classes are given for k = 2, 3, 4, 5 and 6 for the uniform, normal and exponential distributions. Also, the number of groups to use and the "robustness" of the optimal grouping are investigated. It is noted that the "mathematical problem" in determining the optimal classes appears in other studies; a rationale for this is presented. [ABSTRACT FROM AUTHOR]