We study theoretically by means of quantum electrodynamics the vortex radiation a relativistic electron in uniform magnetic field. The exact expressions for probability emission photon with certain angular momentum are found. classical asymptotics ħ→0 this does not match flux density calculated method using symmetrized energy-momentum tensor. Although integrated over directions is same both cas...

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.

A graph G is called randomly H − decomposable if every maximal H − packing in G uses all edges in G. G is called H − equipackable if every maximal H − packing in G is also a maximum H − packing in G. M2 − decomposable graphs, randomly M2 − decomposable graphs and M2 − equipackable graphs have been characterized. The definitions could be generalized to multigraphs. And M2 − decomposable multigra...

The number ofmoves required to solve any configuration of Rubik’s cube has held a fascination for over 25 years. A new upper bound of 26 is produced. More important, a new methodology is described for finding upper bounds. The novelty is two-fold. First, parallel disks are employed. This allows 1.4 × 1012 states representing symmetrized cosets to be enumerated in seven terabytes. Second, a fast...

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 ...