This article concerned on the study of signature submanifolds for curves under Lie group actions SE(2), SA(2) and for surfaces under SE(3). Signature submanifold is a regular submanifold which its coordinate components are differential invariants of an associated manifold under Lie group action, and therefore signature submanifold is a key for solving equivalence problems.

The aim of this work is to introduce and study the nondegenerate inner product $<\cdot , \cdot >_{det}$ induced by determinant map on space $Sym(2)$ symmetric $2\times 2$ real matrices. This bilinear form index $2$ defines a rational function pairs rays in plane an associated $2$-torus can be expressed with usual Hopf bundle projection $S^3\rightarrow S^2(\frac{1}{2})$. Also, trea...

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...