Raymond N. Greenwell

Optimal mutation probability for genetic algorithms

We derive the value of the mutation probability which maximizes the probability that the genetic algorithm finds the optimum value of the objective function under simple assumptions. This value is compared with the optimum mutation probability derived in other studies. An empirical study shows that this value, when used with a larger scaling factor in linear scaling, improves the performance of the genetic algorithm. This feature is then added to a model developed by Hinton and Nowlan which allows certain bits to be guessed in an effort to increase the probability of finding the optimum solution.

On the existence of a reasonable upper bound for the van der waerden numbers

Abstract Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers { x 1 , x 2 , …, x n } either that form an arithmetic sequence or for which there exists a polynomial f ( x ) = Σ i = 0 n − 2 a i x i with a i ϵ Z , a n − 2 > 0, and x j + 1 = f ( x j ). We denote by q ( n ) the least positive integer such that if {1, 2, …, q ( n )} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q ( n ), as well as values of q ( n ) for n ⩽ 5. A stronger upper bound for q ( n ) is conjectured and is shown to imply the existence of a similar bound on the n th van der Waerden number.

Randomized rejection procedure for the two-sample Kolmogorov–Smirnov statistic

Abstract The two-sample Kolmogorov–Smirnov test is unable to achieve an arbitrary probability of Type I error because it can only take on a limited number of discrete values. We offer a randomized procedure that achieves any specified value of α. We derive formulas for approximating the achievable p-values immediately above and below the desired value of α. For the value of the statistic corresponding to the p-value greater than α, our procedure rejects the null hypothesis randomly with probability sufficient to achieve the specified α. Such a procedure is particularly appropriate for simulation studies. Our procedure leads to a different continuity correction than the one proposed by Kim (J. Amer. Statist. Assoc. 64 (1969) 1625), using the criterion that the continuity correction should cause the randomized procedure to reject the null hypothesis with probability α.

Values and bounds for Ramsey numbers associated with polynomial iteration

Abstract Ramsey numbers similar to those of van der Waerden are examined. Rather than considering arithmetic sequences, we look at increasing sequences of positive integers {x1,x2,…,xn} for which there exists a polynomial ⨍(x)=∑ r i =0 a i x i , with aiϵZ and x j +1 =⨍(x j ) . We denote by pr(n) the least positive integer such that if [1,2,…,pr(n)] is 2-colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree ⩽r. We give values for pr(n) for n⩽5, as well as lower bounds for p1(n) and p2(n). We also give an upper bound for certain Ramsey numbers that are in between pn−2(n) and the nth van der Waerden number.

Statistical Significance of Ranking Paradoxes

When nonparametric statistical tests are used to rank-order a list of alternatives, Simpson-like paradoxes arise, in which the individual parts give rise to a common decision, but the aggregate of those parts gives rise to a different decision. Haunsperger (2003) and Bargagliotti (2009) showed that the Kruskal-Wallis (Kruskal and Wallis, 1952), Mann-Whitney (Mann and Whitney, 1947), and Bhapkars V (Bhapkar, 1961) nonparametric statistical tests are subject to these types of paradoxes. We further investigate these ranking paradoxes by showing that when they occur, the differences in rankings are not statistically significant.

Some new bounds and values for van der Waerden-like numbers

Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x1,x2,...,xn} that either form an arithmetic progression or for which there exists a polynomialf with integer coefficients and degree exactlyn − 2, andxj+1 =f(xj). We denote byq(n, k) the least positive integer such that if {1, 2,...,q(n, k)} is partitioned intok classes, then some class must contain a sequence of the type just described. Upper bounds are obtained forq(n, 3), q(n, 4), q(3, k), andq(4, k). A table of several values is also given.

Fluid flow through a partially filled cylinder

The laminar flow of viscous incompressible fluid through a partially filled circular cylinder is studied. The governing equations reduce to Laplaces equation, and the region of solution is the union of two intersecting circles. This region is mapped onto the unit circle by a conformal transformation, so that the problem may be solved by the Poisson integral formula. The integration is performed by a combination of residues and numerical integration. The results agree very well with exact analytic solutions at 50%, 85.4%, and 100% depth to diameter ratios. It is found that by not filling the cylinder completely, the flow rate can be increased by a significant 27.3%.

Combinatorics and Statistical Issues Related to the Kruskal–Wallis Statistic

We explore criteria that data must meet in order for the Kruskal–Wallis test to reject the null hypothesis by computing the number of unique ranked datasets in the balanced case where each of the m alternatives has n observations. We show that the Kruskal–Wallis test tends to be conservative in rejecting the null hypothesis, and we offer a correction that improves its performance. We then compute the number of possible datasets producing unique rank-sums. The most commonly occurring data lead to an uncommonly small set of possible rank-sums. We extend prior findings about row- and column-ordered data structures.

Generalization of results on almost closed compartment systems

Abstract The results of Rubinow [1] on closed or almost closed compartment systems are generalized. It is shown that observation of only one compartment uniquely determines the fractional material transport rates and the compartment sizes provided certain restrictions are fulfilled. The system does not need to be mammillary or catenary, nor does it need to be almost closed.

Partially ordered sets and stratification

We employ partially ordered sets to describe the stratification of a social system, using rank to define the strata. We present a simple method of computing the matrix corresponding to the Hasse diagram and prove its correctness. This methodology is applied to analyze the hierarchy of countries that have won at least one Olympic medal. Four different definitions of dominance are given, leading to four different hierarchies and Hasse diagrams. We also prove that any of these definitions preserve any ordering based on giving different weights to gold, silver, and bronze medals. We study dominance between adjacent strata and note how the system changes with time. We present a case analysis for Poland as an illustration of the set of data that can be computed for any country.