We discuss the algebraic classification of the Weyl tensor in higher dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is obtained by specifying the alignment type and utilizing the notion of reducibility. For a complete classification it is then necessary to count aligned directions, the dimension of the alignment variety, and the multiplicity of principal directions. The present classification reduces to the classical Petrov classification in four dimensions. Some applications are briefly discussed.

Abstract. A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X 1 -Jacobi and X 1 -Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1, 1] or the half-line [0, ∞), respectively, and they are a basis of the corresponding L 2 Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial, then it must be either the X 1 -Jacobi or the X 1 -Laguerre SturmLiouville problem. A Rodrigues-type formula can be derived for both of the X 1 polynomial sequences.

It was argued that the basic principles of operation of human memory can be understood as an optimization to the information-retrieval task that human memory faces. Basically, memory is using the statistics derived from past experience to predict what memories are currently relevant. It was shown that the effects of frequency, recency, and spacing of practice can be predicted from the statistical properties of information use. The effects of memory prompts, cues, and primes can be predicted on the assumption that memory is estimating which knowledge win be needed from past statistics about interitem associations. This analysis was extended to account for fan effects. Memory strategies were analyzed as external to the process of statistical optimization. Memory strategies are attempts to manipulate the statistics of information presentation to influence the optimal solution derived by memory. The classic buffer-rehearsal model for free recall is analyzed as a strategy to manipulate the statistics of information presentation.Human memory is typically viewed by lay people as quite a defective system. For instance, over the years we have participated in many talks with artificial intelligence researchers about the prospects of using human models to guide the development of artificial intelligence programs. Invariably, the remark is made, "Well, of course, we would not want our system to have something so unreliable as human memory." Actual memory researchers seldom comment on the adaptiveness of memory (but see Bjork & Bjork, 1988;Sherry & Schacter, 1987). One seldom finds arguments for a theory of memory mechanisms cast in terms of the adaptiveness of these mechanisms. Rather, the typical argument for a memory mechanism is by reference to its ability to fit the data at hand. The implied inference is that the actual combination of mechanisms is pretty arbitrary.Certainly, this characterization is fairly accurate of our own writings on memory despite our advocacy of the ACT* theory (Anderson, 1983) whose initials stand for "Adaptive Control of Thought".In this article, we argue that human memory is adaptively designed and that we can understand a great deal about memory phenomena by understanding its adaptiveness. The analysis being offered here is not meant to supplant existing mechanistic accounts such as ACT* but to supplement them by showing that they implement a rational memory design. This article begins with a framing of the information-processing problem that memory faces. Then we derive the optimal memory behavior, given this framing. We show that this predicts many of the major results in human memory. These results are very general trends that appear across particular memory paradigms. As a

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.

We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p 1 , p 2 , . . . of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper.

All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.