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This paper studies the situation when two four-dimensional Lorentz manifolds (that is, space–times) admit the same (unparametrised) geodesics, that is, when they are projectively related. A review of some known results is given and then the problem is considered further by treating each holonomy type in turn for the space–time connection. It transpires that all holonomy possibilities can be dealt with completely except the most general one and that the consequences of two space–times being projectively related leads, in many cases, to their associated Levi-Civita connections being identical.

This paper considers 4-dimensional manifolds upon which there is a Lorentz metric, h, and a symmetric connection and which are originally assumed unrelated. It then derives sufficient conditions on the metric and connection (expressed through the curvature tensor) for the connection to be the Levi-Civita connection of some (local) Lorentz metric, g, and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived.

Problem structure, or linkage, refers to the interaction between variables in a black-box fitness function. Discovering structure is a feature of a range of algorithms, including estimation of distribution algorithms (EDAs) and perturbation methods (PMs). The complexity of structure has traditionally been used as a broad measure of problem difficulty, as the computational complexity relates directly to the complexity of structure. The EDA literature describes necessary and unnecessary interactions in terms of the relationship between problem structure and the structure of probabilistic graphical models discovered by the EDA. In this paper we introduce a classification of problems based on monotonicity invariance. We observe that the minimal problem structures for these classes often reveal that significant proportions of detected structures are unnecessary. We perform a complete classification of all functions on 3 bits. We consider nonmonotonicity linkage discovery using perturbation methods and derive a concept of directed ordinal linkage associated to optimization schedules. The resulting refined classification factored out by relabeling, shows a hierarchy of nine directed ordinal linkage classes for all 3-bit functions. We show that this classification allows precise analysis of computational complexity and parallelizability and conclude with a number of suggestions for future work.

Estimation of distribution algorithms (EDAs) use structure learning to build a statistical model of good solutions discovered so far, in an effort to discover better solutions. The non-zero coefficients of the Walsh transform produce a hypergraph representation of structure of a binary fitness function; however, computation of all Walsh coefficients requires exhaustive evaluation of the search space. In this paper, we propose a stochastic method of determining Walsh coefficients for hyperedges contained within the selected subset of the variables (complete local structure). This method also detects parts of hyperedges which cut the boundary of the selected variable set (partial structure), which may be used to incrementally build an approximation of the problem hypergraph.

A region with chaotic magnetic field lines where the magnetic field (B) and plasma velocity (v) are continuous and differentiable and satisfy the dissipative incompressible magnetohydrodynamic equations with magnetic diffusivity η and kinematic viscosity ν is considered. It is proved then that if v×B and (∇×v)×v are potential, the structurally stable solutions describing such chaotic regions are characterized by a decaying linear magnetic force-free field and Beltrami flow of the form B=B0 exp(−α2ηt)b, v=v0 exp(−α2νt)b, where b=b(r) such that ∇×b=αb, ∇⋅b=0 and B0, v0, and α are constants. Purely hydrodynamic flows are a particular case with B0=0. A simple example of a chaotic force-free field is also constructed.

Linear null points of a magnetic field may come together and coalesce at a secondorder null, or vice versa a second-order null may form and split, giving birth to a pair of linear nulls. Such local bifurcations lead to global changes of magnetic topology and in some cases release of magnetic energy. In two dimensions the null points are of X or O type and the flux function is a Hamiltonian; the magnetic field may undergo addle-centre, pitchfork or degenerate resonant bifurcations. In three dimensions the null points and their creation or annihilation by bifurcations are considerably more complex. The nulls possess a skeleton consisting of a spine curve and a fan surface and are of radial-type (proper or improper) or spiral-type; the type of null and the inclination of spine and fan depend on the magnitudes of the current components along and normal to the spine. In cylindrically symmetric fields a comprehensive treatment is given of the various types of saddle-node, Hopf and saddle-node—Hopfbifurcations. In fully three-dimensional situations examples are given of saddle-node and degenerate bifurcations, in which generically two nulls are created or destroyed and are joined by a separator field line, which is the intersection of the two fans. Furthermore, global bifurcations can create chaotic field lines that could perhaps trigger energy release in, for example, solar flares.

Magnetic reconnection has traditionally been associated exclusively with the presence of magnetic null points or field lines tangential to a boundary. However, in many cases introducing a three-dimensional perturbation in a two-and-half-dimensional magnetic configuration implies the disappearance of separatrices. Faced with this structural instability of separatrices when going from two-and-half to three-dimensional configurations, several approaches have been investigated to replace the topological ideas familiar in two-dimensional, but no unanimity has yet emerged on the way reconnection should be defined. While it is true that the field line linkage is continous in three-dimensional, we show here that extremely thin layers (called quasi-separatrix layers (QSLs)) are present. In these layers the gradient of the mapping of field lines from one part of a boundary to another is very much larger than normal (by many orders of magnitude). Even for highly conductive media these extremely thin layers behave physically like separatrices. Thus reconnection without null points can occur in QSLs with a breakdown of ideal MHD and a change in connectivity of plasma elements. We have analyzed several twisted flux tube configurations, going progressively from two-and-half to three-dimensional, showing that QSLs are structurally stable features (in contrast to separatrices). The relative thickness w of QSLs depends mainly on the maximum twist ; typically, with two turns, w≃10 -6 , while with four turns, w≃10 -12 . In these twisted configurations the shape of the QSLs, at the intersection with the lower planar boundary, is typical of the two ribbons observed in two-ribbon solar flares, confirming that the accompanying prominence eruption involves the reconnection of twisted magnetic structures. We conclude that reconnection occurs in three-dimensional in thin layers or QSLs, which generalise the traditional separatrices (related only to magnetic null points or field lines tangential to the boundary).

Projective collineations in spacetimes, i.e. vector fields generating local groups of geodesic-preserving diffeomorphisms, are studied. The situation for Einstein spaces is resolved completely and some general results are established regarding arbitrary spacetimes. Examples of proper projective collineations are constructed.

A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided., Comment: Comments: 23 pages, LaTeX; typos corrected, page 9 last line corrected to $g'=e^{2\chi}a^{-1}$

A study is made of the possible holonomy group types of a space-time for which the energy-momentum tensor corresponds to a null or non-null electromagnetic field, a perfect fluid or a massive scalar field. The case of an Einstein space is also included. The techniques developed are also applied to vacuum and conformally flat space-times and contrasted with already known results in these two cases. Examples are given., Latex, 14 pages, no figure

International audience; This paper presents a study of the situation when two space-times admit the same (unparametrised) geodesics, that is, when they are projectively related. The solution is based on the curvature class and the holonomy type of a space-time and it transpires that all holonomy possibilities can be solved except the most general one and that the consequence of two space-times being projectively related leads, in many cases, to their associated Levi-Civita connections being identical. Some results are also given regarding the general case. It is also shown that the holonomy types of projectively related space-times are very closely related. The theory is then applied, with Einsteins principle of equivalence in mind, to " generic " space-times.

This paper considers the symmetries of the curvature tensor (curvature collineations) and of the Weyl conformal tensor (Weyl conformal collineations) in general relativity. Some general results are reviewed for later application, some new ones proved and many special cases are investigated. Particular emphasis is laid on the interrelations between these two types of symmetries. A number of instructive examples of such symmetries are given.

This paper is concerned with the extent to which the geodesics of space-time (that is, the experimental consequences of the principle of equivalence) determine the metric in general relativity theory and, in particular, in Friedmann–Robertson–Walker–Lemaitre (FRWL) space-times. Thus it discusses projective structure in these space-times. The approach will be from a geometrical point of view and it is shown that if two space-time metrics share the same (unparametrized) geodesics and one is a (generic) FRWL metric then so is the other and that each is a member of a well defined family of projectively related (FRWL) metrics. Similar techniques are then applied to study the existence and properties of symmetries of the Weyl projective tensor and projective symmetries in FRWL space-times.