The repertoire of set operators available in Constructive Solid Geometry (CSG) may be extended by the Minkowski addition and subtraction operators. In this way the problem of visualizing 3D solids deened as a Minkowski addition or subtraction of elementary objects can be solved by using CSG in combination with ray tracing. The method consists in transforming a Minkowski term with multiple opera...

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact f...

In this paper we introduce the Minkowski weighted partition around medoids algorithm (MW-PAM). This extends the popular partition around medoids algorithm (PAM) by automatically assigning K weights to each feature in a dataset, where K is the number of clusters. Our approach utilizes the within-cluster variance of features to calculate the weights and uses the Minkowski metric. We show through ...

The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Al...

This is a supplement to our previous research note [3]. In [3], we gave a characterization of biharmonic curves in Minkowski 3-space. More precisely, we pointed out that every biharmonic curves with nonnull principal normal in Minkowski 3-space is a helix, whose curvature κ and torsion τ satisfy κ2 = τ2. In the classification of biharmonic curves in Minkowski 3-space due to Chen and Ishikawa [1...

Abstract This paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and equivalent Minkowski mixed log-volume.

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra defined in the Fock representation. That representation has central charge c = 1. The Virasoro algebra of operators gets a manifest geometr...

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes called perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of faces of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face coun...

In tolerance analysis, it is necessary to check that the cumulative defect limits specified for the component parts of a product are compliant with the functional requirements expected of the product. Cumulative defect limits can be modelled using a calculated polytope, the result of a set of intersections and Minkowski sums of polytopes. This article presents a method to be used to determine f...

We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about q-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra gl2,q. Our point of view fixes the signature of the m...