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In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris’ procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses. The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied—a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, “non-standard” applications of the Farris transform.

In this paper we study how to invert random functions under different criteria. The motivation for this study is phylogeny reconstruction, since the evolution of biomolecular sequences may be considered as a random function from the set of possible phylogenetic trees to the set of collections of biomolecular sequences of observed species. Our results may affect how we think about maximum likelihood estimation (MLE) in phylogeny. For inverting random functions, MLE is optimal under a first criterion, although it is not optimal under a second criterion which is at least equally natural but more conservative. Furthemore, MLE has to be used differently from the way it has been used in the phylogeny literature, if we have a prior distribution on trees and mutation mechanisms and want to keep MLE optimal under the same first criterion. Some of the results of this paper have been known in the setting of statistical decision theory, but have never been discussed in the context of phylogeny.

In this paper, we present an algorithm which can prove algebraic relations involving $${\eta}$$ -quotients, where $${\eta}$$ is the Dedekind eta function.

For a k-connected matroid M, Clark andWhittle showed there is a tree that displays, up to a natural equivalence, all non-trivial k-separations of M. In this paper, we present an algorithm for constructing such a tree, and prove that, provided the rank of any subset of E(M) can be found in constant time, the algorithm runs in polynomial time in |E(M)|.

Calculating the rooted subtree prune and regraft (rSPR) distance between two rooted binary phylogenetic trees is a frequently applied process in various areas of molecular evolution. However, computing this distance is an NP-hard problem and practical algorithms for computing it exactly are rare. In this paper, a divide-and-conquer approach to calculating the rSPR distance is established. This approach breaks the problem instance into a number of smaller and more tractable subproblems. Two reduction rules which were previously used to show that computing the rSPR distance is fixed-parameter tractable can easily be used to complement this new theoretical result, and so a significant positive impact on the running time of calculating this distance in practice is likely.

In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion, that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find new weighing matrices of order 2n and weights 2n – 13, 2n – 17 constructed from two circulants.

In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. By developing the Omega package we have shown that Partition Analysis is ideally suited for being supplemented by computer algebra methods. The object of this paper is to present a significant algorithmic improvement of this package. It overcomes a problem related to the computational treatment of roots of unity. Moreover, this new reduction strategy turns out to be superior to "The Method of Elliott" which is described in MacMahon's book. In order to make this article as self-contained as possible we give a brief introduction to Partition Analysis together with a variety of illustrative examples. For instance, the generating function of magic pentagrams is computed.