Sphere Packings, Lattices and Groups

A euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges

“…Similar to the four dimensional example, we introduce four three dimensional vectors to relate the point n to a spacetime point. These lattice vectors can be chosen as The matrix e m · e n is the Gram matrix of A * 3 [30], also known as body-centered cubic (bcc) lattice.…”

Many-to-Many Feature Matching Using Spherical Coding of Directed Graphs

“…A spherical code is the distribution of a finite set of n points on the surface of a unit sphere such that the minimum distance between any pair of points is maximized [6]. Equivalently, one can try to minimize the radius r of a d-dimensional sphere such that n points can be placed on the surface, where any two of the points are at angular distance 2 from each other.…”

“…Finally, to complete the embedding, we translate the subcones away from the origin along their directional rays to positions defined by the path lengths in the tree. , and the minimum angle between two vectors to be 2/r provides us with L well-separated vectors [6]. In Figure 4, we have 4 such vectors emanating from the origin.…”

Classification of integral lattices with large class number

“…The masses are calculated using the formulas and tables given in [5]. The sign occurring in the genus symbol for the prime which is needed for this calculation is determined from the existence conditions of [4], Chapter 15, Theorem 13. For = 3 and n = 12 (genus of the Coxeter-Todd lattice), the classification had been obtained previously in [16], using a different method (root systems, glue codes, the mass formula).…”

“…We refer to [9], Chap. 10 and [4], Chap. 15 for general results about integral lattices (or, equivalently, integral quadratic forms) and their classification, and to [1], [11], [15], [16], [19] for more specialized investigations close to the subject of the present paper.…”