In this paper, we develop a distributed algorithm to localize a network of robots moving arbitrarily in a bounded region. In the case of such mobile networks, the main challenge is that the robots may not be able to find nearby robots to implement a distributed algorithm. We address this issue by providing an opportunistic algorithm that only implements a location update when there are nearby r...

We study the barycentric straightening of simplices in higher rank irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r ≥ 2 (excluding S L(3,R)/S O(3) and S L(4,R)/S O(4)), the p-Jacobian has uniformly bounded norm, provided p ≥ n − r + 2. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group G, in deg...

Theorem 1 (Line). The equation of a line is ux+ vy + wz = 0 where u, v, w are reals. (These u, v and w are unique up to scaling.) ∗Mewto55555, Missouri. I can be contacted at [email protected]. †v Enhance, SFBA. I can be reached at [email protected]. 1For ABC counterclockwise, this is positive when P , Q and R are in counterclockwise order, and negative otherwise. When ABC is la...

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on posets.

This paper introduces a method for defining and efficiently computing barycentric coordinates with respect to polygons on general surfaces. Our construction is geared towards injective polygons (polygons that can be enclosed in a metric ball of an appropriate size) and is based on replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, and gene...

Let G be an abelian group with n elements. Let S be a sequence of elements of G, where the repetition of elements is allowed. Let |S| be the cardinality, or the length of S. A sequence S ⊆ G with |S| ≥ 2 is barycentric or has a barycentric-sum if it contains one element aj such that ∑ ai∈S ai = |S|aj . This paper is a survey on the following three barycentric constants: the k-barycentric Olson ...

ÐWe present some very efficient and accurate methods for computing tangent curves for three-dimensional flows. Our methods work directly in physical coordinates, eliminating the usual need to switch back and forth with computational coordinates. Unlike conventional methods, such as Runge-Kutta, for computing tangent curves which give only approximations, our methods produce exact values based u...

Barycentric plotting, achieved by placing gaussian kernels in distant corners of the feature space and projecting multidimensional output of neural network on a plane, provides information about the process of training and certain features of the network. Additional visual guides added to the plot show tendencies and irregularities in the training process.

In this paper, (d + 1)-pencil lattices on simplicial partitions in Rd are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact l...

The paper presents the solution of a three-dimensional (3-D) magnetostatic problem using the finite formulation of an electromagnetic field. Two different approaches, based on a primal-dual barycentric discretization of the 3-D space, are presented, considering as unknowns either the magnetic fluxes or the circulations of the vector potential. Results on simple reference configurations are repo...