Linear Independent Solutions Research Articles

The unique solution of projective invariants of six points from four uncalibrated images

In this paper, we show that four uncalibrated images are sufficient to uniquely determine the 3D projective invariants of a set of six points in general position in space. An algorithm for computing the unique solution is proposed. It computes by solving a set of linear equations for its two linear independent solutions and is simpler than other algorithms which compute the three possible solutions of 3D projective invariants by solving a set of nonlinear equations. However, a fourth image is needed to get the linear unique solution. Finally, experimental results have shown the feasibility of this algorithm.

Inhomogeneous waves in transversely isotropic solids

Wave propagation in transversely isotropic, uniform, linear, viscoelastic solids is investigated by studying the evolution along the symmetry axis of the stress-displacement vector as modelled by a system of six ordinary differential equations. Families of six linearly-independent solutions are determined which span the general integral. The independent generators are inhomogeneous waves and are parametrized by the component of the wave vector in the plane of isotropy. The complex valuedness of the wave vector makes the problem three-dimensional. Horizontally and vertically polarized waves are then characterized. The wave solutions are obtained also when the wave propagation is governed by a non-semisimple matrix. As an application, polynomial equations are derived for the determination of free-surface waves and Stoneley waves.

Algorithm for the estimation of projective invariants of six points from uncalibrated images

A new algorithm for the estimation of 3-D projective invariants from uncalibrated images is proposed. It solves the uniqueness of the estimation of 3-D projective invariants of six points from uncalibrated images by introducing a 4th image. It also estimates the 3-D projective invariants by using two linear independent solutions of linear equations and is simpler than other algorithms which estimate the 3-D projective invariants by solving the nonlinear equations.

NON-CONSERVATIVE INSTABILITY OF A TIMOSHENKO BEAM SUBJECTED TO A PARTIALLY TANGENTIAL FOLLOWER FORCE

The governing differential equations and boundary conditions for the non-conservative instability of a Timoshenko beam subjected to an end partial tangential follower force are derived via Hamilton's principle. The two coupled governing differential equations are reduced to one fourth order ordinary differential equation in terms of the flexural displacement. The characteristic equation is expressed in terms of four linear independent fundamental solutions of the system. The influences of the tangency coefficient, the slenderness ratio and the elastically restrained boundary conditions on the elastic instability and the critical load of a Timoshenko beam are investigated. The boundary curves for the flutter and divergence instability of clamped–elastically restrained beams are determined.

Hyperanalytic riemann boundary value problems on rectifiable closed curves

We extend the results ot [3] for the Riemann boundary value problem of a hyperanalytic functions to the case of data given on rectifiable closed curves. An explicit form of the general solution is constructed. The necessary and sufficient conditions for solvability of the boundary value problem, and the relations between the index and the number of linear independent solutions are given.

An exact solution in the dynamical diffraction theory of lamellarly distorted crystals

An exact solution of dynamical diffraction was found for a lamellarly distorted infinite crystal. Here the deviation of the lattice spacing was assumed to have the form D0 tanh αx where the spatial variable x is normal to the net plane concerned. The problem is reduced to solving an ordinary differential equation of second order. The linearly independent solutions (LIS) are represented in a very symmetric manner with the use of the U function defined here. They are essentially hypergeometric functions with three complex parameters which are specified in terms of the diffraction condition and the lattice distortion. The analytical properties of the LIS's and their physical interpretation are described. Rocking curves are calculated for both the Darwin and Ewald cases. The theory can be applied to a variety of monotonic lattice distortions.

Series expansions and small solutions for Volterra equations of convolution type

In general, retarded functional differential equations have an infinite dimensional character, in the sense that there exist an infinite number of linear independent characteristic solutions p j ( t) e λ j t , where λ j denotes a zero of a transcendental equation and p j a polynomial. In this paper we use Laplace transform methods to study the asymptotic behaviour of the solutions of this type of differential equations. Furthermore, we present necessary conditions such that a solution can be represented as a series of characteristic solutions. With these results we then can study the geometric structure of the strongly continuous semigroup T( t) associated with a retarded functional differential equation. The main result will be a characterization of the closure of the system of generalized eigenfunctions of the infinitesimal generator A of T( t).

New methods for the solution of the coupled equations describing inelastic scattering in the presence of Coulomb interaction

Inelastic heavy-ion scattering is described by a set of coupled homogeneous second-order differential equations. In the canonical method the boundary value problem corresponding to the scattering wave solutions had to be solved by calculating a complete set of linear independent solutions and choosing the appropriate linear combination. We now reformulate the Schrödinger equations for the radial wave functions, obtaining first-order differential equations which can be solved by iteration. Only few iterations of the coupled equations are then needed. A discussion of the accuracy and feasibility of two different iteration methods is given.