The hypermetric cone Hn is the cone in the space R n(n1)/2 of all vectors d = (dij) 1 <i< j <_ n satisfying the hypermetric inequalities: El<_i<j<n zjzjdij ~0 for all integer vectors z in Z n with E l < i < n zi = 1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres and L-polytopes in lattices). As an application, we show that the hyp...

The neighbour joining method is a distance based method for constructing evolutionary trees [2]. Conceptually, it starts out with a star-formed graph where each leaf corresponds to a species, and iteratively picks two nodes adjacent to the root and joins them, by inserting a new node between the root and the two selected nodes. When joining nodes, the method selects the node pair i; j that mini...

In multimodal function optimization, niching techniques create diversification within the population, thus encouraging heterogeneous convergence. The key to the effective diversification is to identify the similarity among individuals. Without knowledge of the fitness landscape, it is usually determined by uninformative assumptions. In this article, we propose a method to estimate the sharing d...

The distance matrix of a simple connected graph G is D(G) = (dij), where dij is the distance between ith and jth vertices of G. The multiset of all eigenvalues of D(G) is known as the distance spectrum of G. Lin et al.(On the distance spectrum of graphs. Linear Algebra Appl., 439:1662-1669, 2013) asked for existence of graphs other than strongly regular graphs and some complete k-partite graphs...

A recent paper by Sammon' describes a useful technique for analyzing data and helping find clusters. It consists of a nonlinear mapping of the original data into two dimensions (for visual inspection) which is superior for many purposes to use of the first two principal components. The present correspondence points out that essentially the same mapping can be achieved by a widely used, easily o...

In the all-pair shortest distance problem, one computes the matrix D = (dij) where d. . iJ is the minimum weighted length of any path from vertex i to vertex j in a directed complete graph with a weight on each edge. In all the known algorithms, a shortest path p.. 13 achieving dij is also implicitly computed. In fact, log3 f(n) is an informationtheoretic lower bound where f(n) is the total num...

We construct an infinite dimensional quasi-polyadic equality algebra A such that its cylindric reduct is representable, while A itself is not representable. 1 The most well known generic examples of algebraizations of first order logic are Tarski’s cylindric algebras (CA) and Halmos’ polyadic equality algebras (PEA). The theory of cylindric algebras is well developed in the treatise [10], [11],...

Duration Calculus (DC) is a real-time logic with measurement of duration of propositions in observation intervals. It is a highly expressive logic with continuous time behaviours (also called signals) as its models. Validity checking of DC is undecidable. We propose a method for validity checking of Duration Calculus by reduction to a sampled time version of this logic called well sampled Inter...