Background: Adolescent idiopathic scoliosis is characterized by a three-dimensional deviation of the vertebral column and its etiopathogenesis is unknown. Various factors cause idiopathic scoliosis, and among these a prominent role has been attributed to the vestibular system. While the deficits in sensorimotor transformations have been documented in idiopathic scoliosis patients, little attention has been devoted to their capacity to integrate vestibular information for cognitive processing for space perception. Seated idiopathic scoliosis patients and control subjects experienced rotations of different directions and amplitudes in the dark and produced saccades that would reproduce their perceived spatial characteristics of the rotations (vestibular condition). We also controlled for possible alteration of the oculomotor and vestibular systems by measuring the subject's accuracy in producing saccades towards memorized peripheral targets in absence of body rotation and the gain of their vestibulo-ocular reflex.

A group analysis of a system describing an ideal plastic flow is made in order to obtain analytical solutions. The complete Lie algebra of point symmetries of this system are given. Two of the infinitesimal generators that span the Lie algebra are original to this paper. A classification into conjugacy classes of all one- and two-dimensional subalgebras is performed. Invariant and partially invariant solutions corresponding to certain conjugacy classes are obtained using the symmetry reduction method. Solutions of algebraic, trigonometric, inverse trigonometric and elliptic type are provided as illustrations and other solutions expressed in terms of one or two arbitrary functions have also been found. For some of these solutions, a physical interpretation allows one to determine the shape of feasible extrusion dies corresponding to these solutions. The corresponding tools could be used to curve rods or slabs, or to shape a ring in an ideal plastic material by an extrusion process.

In this paper, we study the Lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions. The infinitesimal generators that span the Lie algebra for this system are obtained. We completely classify the subalgebras of up to codimension two in conjugacy classes under the action of the symmetry group. Based on invariant forms, we use Ansatzes to compute symmetry reductions in such a way that the obtained solutions cover simultaneously many invariant and partially invariant solutions. We calculate solutions of the algebraic, trigonometric, inverse trigonometric and elliptic type. Some solutions depending on one or two arbitrary functions of one variable have also been found. In some cases, the shape of a potentially feasible extrusion die corresponding to the solution is deduced. These tools could be used to thin, curve, undulate or shape a ring in an ideal plastic material

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraicgeometric point of view, which allows the introduction of specific firstorder side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems, based on the introduction of specific rotation matrices, enables us to weaken the integrability condition. It allows us to establish a connection between the structure of the set of integral elements and the possibility of constructing specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions are obtained in explicit form, including the general integral for the latter system of equations.

We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous hyperbolic and elliptic systems is formulated. The algebraization of these systems enables us to construct certain classes of solutions for which the matrix of derivatives of the unknown functions is expressible in terms of special orthogonal matrices. These solutions can be interpreted as nonlinear superpositions of k waves (or k modes) in the case of hyperbolic (or elliptic) systems, respectively. Theoretical considerations are illustrated by several examples of inhomogeneous hydrodynamic-type equations which allow us to construct solitonlike solutions (bump and kinks) and multiwave (mode) solutions.