Advances in Mathematics Research presents original studies on the leading edge of mathematics. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. Topics discussed include Blackwell's ordering on the set of monotone transition matrices; symplectic polynomial invariants; new results concerning the Bernstein-Nikol'skii inequality theory; approximation theory by neural networks; torsion fields, quantum geometrics, Brownian motions and statistical thermodynamics and the harmonic analysis of finite groups, number theory and efficient quantum cryptographic algorithms.