This book focuses on advances in mathematics research. Chapter 1, derives and discusses features of probability distributions connected to the motion of substances in a channel of network. Chapter 2 analyzes control methods like Melnikov analysis (MA) techniques which allow for improvements and the development of a theoretical approach to chaos suppression in damped driven systems. Chapter 3 focuses on the issue of solving strongly nonlinear oscillators by applying averaging methods. Chapter 4 considers using the structural-parametric approach (SPA) to describe hysteresis, which removes the difficulties of using known methods of BWH identification. Chapter 5 deals with the use of modal and fuzzy logic for analyzing the content validity of indicators. In Chapter 6 the probability and information entanglements in molecular communication channels are approached and quantum correlations between separated fragments originating from a common molecular "ancestor" state are commented upon. Chapter 7 studies the meshless radial basis function methods forfinance pdes and a meshfree numerical method is proposed for pricing under the CEV model. Chapter 8 proposes a solution to 3D Navier U Stokes equations with the velocity vector and pressure as main unknowns when boundary and initial conditions are specified. Chapter 9 presents asymptotic estimates of eigenvalues for a boundary value problem includes symmetric double well potential and quadratic spectral parameter in a boundary condition. In Chapter 10 quantum dynamics of the modulus and phase components of molecular electronic states are addressed and the associated continuities of the state probability and current descriptors are reexamined. Finally, Chapter 11 examines piezo actuators and the harmonious linearization of the hysteresis characteristic that is applied to calculate the nanosystem with a piezo actuator.
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