"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 multi-parts geometry-based finite elements modelling in biomechanics; categorical abstract algebraic logic; parallelisation of neural network building and training; over-determined boundary value problems for linear equations of elastodynamics and their applications to elastic wave diffraction theory and two finite methods for nearly incompressible linear elasticity using simplicial meshes.