we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

1 Dot product of R The inner product or dot product of R is a function 〈 , 〉 defined by 〈u,v〉 = a1b1 + a2b2 + · · ·+ anbn for u = [a1, a2, . . . , an] , v = [b1, b2, . . . , bn] ∈ R. The inner product 〈 , 〉 satisfies the following properties: (1) Linearity: 〈au + bv,w〉 = a〈u,w〉+ b〈v,w〉. (2) Symmetric Property: 〈u,v〉 = 〈v,u〉. (3) Positive Definite Property: For any u ∈ V , 〈u,u〉 ≥ 0; and 〈u,u〉 =...

This is the second part of the paper (the rst part is published in Journal of AMS 9 1135. In the rst part, we deened for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space Hom(L X i X i ; U) is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group U q sl n...

This article addresses the problem of cardinality estimation in inner product spaces. Given a set high-dimensional vectors, query, and threshold, this estimates number vectors such that their products with query are not less than threshold. is an important for recent machine-learning applications maintain objects, as users items, by using matrices. The requirements solutions high efficiency acc...