Spherical coordinates are a natural orthogonal system to describe wavefronts emanating from a point source. While a regular grid distribution in the Cartesian coordinate system tends to undersample the wavefront description near the source (the highest wavefront curvature) and oversample it away from the source, spherical coordinates, in general, provide a more balanced grid distribution to cha...

The fundamental structure of the universe is posited to be a network of causal relationships. Coordinate systems are interpreted as a regular structure of causal links. The discrete nature of such coordinate systems and the associated aliasing gives rise to the existence of a phase factor. This in turn leads to an interpretation of the probabilistic nature of observation and the path integral a...

We show that the regularity of the gravitational metric tensor cannot be lifted from C to C by any C coordinate transformation in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel’s Theorem [6] which states that such coordinate transformations always...

This paper proves the Commuting Derivations Conjecture in dimension three: if D1 and D2 are two locally nilpotent derivations which are linearly independent and satisfy [D1, D2] = 0 then the intersection of the kernels, A1 ∩ A2 equals C[f ] where f is a coordinate. As a consequence, it is shown that p(X)Y + Q(X, Z, T ) is a coordinate if and only if Q(a, Z, T ) is a coordinate for every zero a ...

We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized coordinate transformations fails to associate. Moreover, in dimensional reduction, the O(d, d) T-duality transformations of fields can be obtained as generalized ...

With increasingly "big" data available in biomedical research, deriving accurate and reproducible biology knowledge from such big data imposes enormous computational challenges. In this paper, motivated by recently developed stochastic block coordinate algorithms, we propose a highly scalable randomized block coordinate Frank-Wolfe algorithm for convex optimization with general compact convex c...

Many large-scale Internet applications optimize their overlay network to reduce latencies. Embedding coordinate systems like Vivaldi are valuable tools for this new range of applications since they propose light-weight algorithms that permit to estimate the latency between any pair of nodes without having to contact them first. It has been recently demonstrated that coordinate systems in genera...

The new CCP4 Coordinate Library is a development aiming to provide a common layer of coordinate-related functionality to the existing applications in the CCP4 suite, as well as a variety of tools that can simplify the design of new applications where they relate to atomic coordinates. The Library comprises a wide spectrum of useful functions, ranging from parsing coordinate formats and elementa...