Matrix completion problems are concerned with determining whether partially speciied matrices can be completed to fully speciied matrices satisfying certain prescribed properties. In this article we survey some results and provide references about these problems for the following matrix properties: positive semideenite matrices, Euclidean distance matrices, completely positive matrices, contrac...

The point placement problem is to determine the positions of a linear set of points, P = {p1, p2, p3, . . . , pn}, uniquely, up to translation and reflection, from the fewest possible distance queries between pairs of points. Each distance query corresponds to an edge in a graph, called point placement graph (ppg), whose vertex set is P . The uniqueness requirement of the placement translates t...

Abstract—We propose an Euclidean geometric representation for the classical detection theory. The proposed representation is so generic that can be employed to almost all communication problems. The hypotheses and observations are mapped into R in such a way that a posteriori probability of an hypothesis given an observation decreases exponentially with the square of the Euclidean distance betw...

Let M be a complete Riemannian manifold with a free cocompact Z-action. Let k(t, m1, m2) be the heat kernel on M . We compute the asymptotics of k(t, m1, m2) in the limit in which t → ∞ and d(m1, m2) ∼ √ t. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on R coming from the Hodge inner product on H(M/Z;R).

The Cayley-Dickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as two-dimensional vectors equipped with a multiplication. The construction can be used to construct, not only the two-dimensional Complex Numbers, but also the four-dimensional Quaternions and the eight-dimensional Octonions. Each...

Experimental results of applying two similarity measurements, Euclidean distance and chord distanc, to test a set of six Brodatz’s textures are reported. Experiments show that in addition to feature extraction, A similarity measurement between images should be simultaneously considered, We also review some other similarity measurements.

Article history: Received 22 May 2014 Accepted 8 November 2014 Available online 25 November 2014 Submitted by J.M. Landsberg MSC: 15B99 13P25 20G20

We address the chromatic number of a class of Euclidean distance graphs having vertex set Q, the 3-dimensional rational space. It is shown that if s is any positive integer with no prime factors congruent to 2 (mod 3), and G is the graph with vertex set Q where any two vertices are adjacent if and only if they are distance √ 2s apart, then G has chromatic number 4. Along the way, we obtain a fe...

a) Develop a new ‘centre of gravity’-based normalisation procedure which better enables visual comparisons of vowel formant plots from different sources, and avoids some of the disadvantages of more well-known procedures (Watt and Fabricius 2003). b) Define geometrical measures of the configurational relations between average vowel positions on formant plots to aid the replicable interpretation...

Let C be a plane convex body. For arbitrary points , denote by , n a b E  ab the Euclidean length of the line-segment . Let be a longest chord of C parallel to the line-segment . The relative distance between the points and is the ratio of the Euclidean distance between and b to the half of the Euclidean distance between and . In this note we prove the triangle inequality in with the relative ...