The triangle removal lemma states that a simple graph with o(n3) triangles can be made triangle-free by removing o(n2) edges. It is natural to ask if this widely used result can be extended to multi-graphs. In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only n2+o(1) triangles that are still far from being triangle-free. On the ot...

While Bregman divergences [3] have been used for several machine learning problems in recent years, the facts that they are asymmetric and does not satisfy triangle inequality have been a major limitation. In this paper, we investigate the relationship between two families of symmetrized Bregman divergences and metrics, which satisfy the triangle inequality. Further, we investigate kmeans-type ...

Laughlin states have recently been constructed on fractal lattices and shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On Sierpinski triangle, for instance, entanglement entropy shows oscillations as a function particle number does not obey area law despite being topologically ordered, density is non-uniform bulk. He...

In this paper, we study the relationship between the Asymmetric Traveling Salesman Problem (ATSP) and the Cycle Cover Problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance, G = (V, E). The strength of the triangle inequality is captured by parametrizing the triangle inequality as follows. A complete directed graph G = (V, E) ...

We present several new quadrature formulas in the triangle for exact integrationof polynomials. The points were computed numerically with a cardinal function algorithm whichimposes that the number of quadrature points N be equal to the dimension of a lower dimensionalpolynomial space. Quadrature forumulas are presented for up to degree d = 25, all which havepositive weights and ...

We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure. MSC 2010. P...

The intersection between 3D objects plays a prominent role in spatial reasoning, geometric modeling and computer vision. Detection of possible intersection between objects can be based on the objects’ triangulated boundaries, leading to computing triangle-triangle intersection. Traditionally there are separate algorithms for cross intersection and coplanar intersection. There is no single algor...

We present five new cubature formula in the triangle and square for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which does not impose any symmetry requirements on the points. Cubature formula are presented which integrate degrees 10, 11 and 12 in the triangle and degrees 10 and 12 in the square. They have positive weights, contain no...

We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Speciically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function (...