In this paper, block distance matrices are introduced. Suppose F is a square block matrix in which each block is a symmetric matrix of some given order. If F is positive semidefinite, the block distance matrix D is defined as a matrix whose (i, j)-block is given by D ij = F ii +F jj −2F ij. When each block in F is 1 × 1 (i.e., a real number), D is a usual Euclidean distance matrix. Many interes...

Self-organizing maps have been adopted in many fields as the data visualization method of choice. The unified distance matrix is the de facto standard for evaluating and interpreting self-organizing maps. In large, high-dimensional problems clusters can be difficult to identify in the plain unified distance matrix. Here we introduce an enhanced version of the unified distance matrix in which cl...

We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain certain results for a perturbat...

Elaeagnus angustifolia L. is a Eurasian tree that has become naturalized and has various ecological, medicinal and economical uses. In this study, a combination of morphological traits and RAPD markers were used to study the presence or absence of an association between genetic variation and morphological features among five populations of E. angustifolia collected from the East Azarbaijan of I...

In this study, we propose a three-stage weighted sum method for identifying the group ranks of alternatives. In the first stage, a rank matrix, similar to the cross-efficiency matrix, is obtained by computing the individual rank position of each alternative based on importance weights. In the second stage, a secondary goal is defined to limit the vector of weights since the vector of weights ob...

Euclidean distance matrix completion problem aims at reconstructing the low dimensional geometry structure of nodes given only a few pairwise distances between nodes [1]. This problem arises in many applications including networks and machine learning where much information of points is laking. For instance, in sensor networks, due to the constraints of energy and communication radius, sensor n...

We investigate relations between (+, ) vector products for binary integer functions . We show that there exists a broad class of products that are equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include, but are not exhausted by: the dominance product, the threshold product and `2p+1 distances for constant p. Our result has the following consequen...

INTRODUCTION The Theory of Distance Geometry [1] allows coordinate-free formulations for most position analysis problems. Using such formulations, the inverse kinematics of a serial robot can be performed by using a distance matrix, one whose entries are squared distances between pairs of points selected on the axes of the robot. While some of these distances are known (such as the distances be...

An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntain...