We’ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n,R) is the group of n by n orthogonal matrices (the group preserving the standard inner product on R). This group has two components, with the component of the identity SO(n,R), the orthogonal matrices of determinant 1. We’ll mostly restrict attention to SO(n,R). An importan...

The concept of a linear ordered orthogonal array is introduced, and its equivalent geometrical configuration is determined when its strength is 3 and 4. Existence of such geometrical configurations is investigated. They are also useful in the study of (T, M, S)-nets.

The problem of synchronization over a group $\mathcal{G}$ aims to estimate collection elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations subset all pairwise ratios the form $G^*_i {G^*_j}^{-1}$. Such has gained much attention recently and finds many applications across wide range scientific engineering areas. In this paper, we consider class problems in which is closed s...

Let $(A,\sigma)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,\sigma)$ contains elements of reduced norm $-1$ if and only the Brauer class $A$ trivial. We generalize this statement to Azumaya algebras involution over semilocal rings, show "if" part fails one allows base ring arbitrary.