Narrow Results

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-) linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds M a conceptual method to rewrite these systems as systems of the form , where is a linear connection on some vector bundle V over M and C : V → T* M ⊗ V is a (vector) bundle map. In particular, if the overdetermined system is linear, will be a linear connection on V and hence the dimension of its solution space is bounded by the rank of V. We will see that the rank of V can be easily computed using representation theory.

The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to prove Valiant's conjecture $VP\ne VNP$. In this paper, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as $GL(V)$-modules.

Within the framework of metric-affine gravity (MAG, metric and an independent linear connection constitute space–time), we find, for a specific gravitational Lagrangian and by using prolongation techniques, a stationary axially symmetric exact solution of the vacuum field equations. This black hole solution embodies a Kerr–de Sitter metric and the post-Riemannian structures of torsion and nonmetricity. The solution is characterized by mass, angular momentum, and shear charge, the latter of which is a measure for violating Lorentz invariance.

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the theory of 𝒟-fields has a model companion 𝒟-CF₀ which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods, can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns out to be particularly fruitful. We propose a modified multigrid V cycle scheme that is not only much simpler, but also more efficient than the standard V cycle. Whereas in the traditional V cycle the residue is passed to the coarser grid levels, this new scheme does not require the calculation of a residue. Instead it works with copies of the charge density on the different grid levels that were obtained from the underlying charge density on the finest grid by wavelet transformations. This scheme is not limited to the pure wavelet setting, where it is faster than the preconditioned conjugate gradient method, but equally well applicable for finite difference discretizations.

The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the equations. As a consequence of studying the prolongation structure of each equation, a Lax pair and Bäcklund transformation are obtained.

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any specific equation. An integrability condition is provided which gives by construction the equation to be investigated and whose components involve the structure constants of an SU(2) Lie algebra. Along side this, a related, different kind of prolongation, a type of Wahlquist–Estabrook prolongation, over a closed differential ideal is discussed and some applications are given.