On the problem of algebraic completeness for the invariants of the Riemann Tensor: I

We present a new determining set, CZ, of Riemann invariants which possesses the minimum degree property. From an analysis on the possible independence of CZ, we are led to the division of all space-times into two distinct, invariantly characterized, classes: a general class M_G+, and a special, singular class M_S For each class, we provide an independent set of invariants (I_G+) ⊂ CZ and I_S ⊂ CZ, respectively) which, with the results of a sequel paper, will be shown to be algebraically complete.

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

We study the set CZ of invariants [Zakhary and Carminati, J. Math. Phys. 42, 1474 (2001)] for the class of space-times whose Ricci tensors possess a null eigenvector. We show that all cases are maximally backsolvable, in terms of sets of invariants from CZ, but that some cases are not completely backsolvable and these all possess an alignment between an eigenvector of the Ricci tensor with a repeated principal null vector of the Weyl tensor. We provide algebraically complete sets for each canonically different space-time and hence conclude with these results and those of a previous article [Carminati, Zakhary, and McLenaghan, J. Math. Phys. 43, 492 (2002)] that the CZ set is determining or maximal.

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.

Zeroes of Riemann's zeta function on the critical line with 20000 decimal digits accuracy

The first 12000 zeroes of Riemann's zeta function on the critical line with 20000 decimal digits accuracy. Format: the zeroes are in text file listed consecutively in decimal representation, each zero starts on a new line.

Zeroes of zeta function presented in this file were calculated on MASSIVE cluster (www.massive.org.au) using Python and packages MPmath version 0.17 and gmpy version 2.1, with a Newton based algorithm proposed by Fredrik Johansson with precision set to 20000 decimal digits. Partial recalculation with higher precision didn't show any loss of accuracy so we expect that the values are correct up to, possibly, a few last digits. We express our thanks to Fredrik Johansson for this algorithm and for development of MPmath as well.

Zeroes of Riemann's zeta function on the critical line with 40000 decimal digits accuracy

The project aims at computing Riemann's zeroes with high accuracy through an analysis of large determinants using Matyasievich's Artless method. Location of Riemann's zeroes is the famous 8th Hilbert problem, and one of Clay's Institute millennium problems.