Determination of all syzygies for the dependent polynomial invariants of the Riemann Tensor. III. mixed invariants of arbitrary degree in the Ricci Spinor

In this paper, we rigorously prove that the complete set of Riemann tensor invariants given by Sneddon [J. Math. Phys. 40, 5905 (1999)] is both minimal and complete. Furthermore, we provide a two-stage algorithm for the explicit construction of polynomial syzygies relating any dependent Riemann tensor invariant to members of the complete set.

On the problem of algebraic completeness for the invariants of the Riemann tensor. II

We study the set of invariants CZ [E. Zakhary and J. Carminati, J. Math. Phys. 42, 1474 (2001)] for the class of space-times whose Ricci tensors do not possess a null eigenvector. We show that all cases are completely backsolvable in terms of sets of invariants from CZ. We provide algebraically complete sets for each canonically different space-time.

Syzygies of the polynomial invariants of the Riemann tensor

This thesis introduces a novel way of writing polynomial invariants as network graphs, and applies this diagrammatic notation scheme, in conjunction with graph theory, to derive algorithms for constructing relationships (syzygies) between different invariants. These algorithms give rise to a constructive solution of a longstanding classical problem in invariant theory.

Colour invariants for machine face recognition

Illumination invariance remains the most researched, yet the most challenging aspect of automatic face recognition. In this paper we investigate the discriminative power of colour-based invariants in the presence of large illumination changes between training and test data, when appearance changes due to cast shadows and non-Lambertian effects are significant. Specifically, there are three main contributions: (i) we employ a more sophisticated photometric model of the camera and show how its parameters can be estimated, (ii) we derive several novel colour-based face invariants, and (iii) on a large database of video sequences we examine and evaluate the largest number of colour-based representations in the literature. Our results suggest that colour invariants do have a substantial discriminative power which may increase the robustness and accuracy of recognition from low resolution images.

Construction of time-dependent dynamical invariants: A new approach

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Closed expressions for Lie algebra invariants and finite transformations

A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements requires the use of projectors, whose coefficients are invariant polynomials. The detailed general forms of these projectors are given. Closed expressions for finite Lorentz transformations, both homogeneous and inhomogeneous, as well as for Galilei transformations, are found as examples.

Darboux invariants for planar polynomial differential systems having an invariant conic

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function circle dot with finitely many singularities on the unit circle T; that is, the continuous mappings g : T -> T such that circle dot o g = circle dot on T. These mappings form a group under composition.