An Affine String Vertex Operator Construction at Arbitrary Level

An affine vertex operator construction at arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of DDF ``oscillators'' and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac-Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac-Moody algebras. A novel interpretation of the affine Weyl group as the ``dimensional null reduction'' of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts.Comment: 15 pages, LaTeX2e, packages amsfonts, amssymb, xspace; final version to appear in J. Math. Phy

Optimal maintenance of multi-component systems: a review

In this article we give an overview of the literature on multi-component maintenance optimization. We focus on work appearing since the 1991 survey "A survey of maintenance models for multi-unit systems" by Cho and Parlar. This paper builds forth on the review article by Dekker et al. (1996), which focusses on economic dependence, and the survey of maintenance policies by Wang (2002), in which some group maintenance and some opportunistic maintenance policies are considered. Our classification scheme is primarily based on the dependence between components (stochastic, structural or economic). Next, we also classify the papers on the basis of the planning aspect (short-term vs long-term), the grouping of maintenance activities (either grouping preventive or corrective maintenance, or opportunistic grouping) and the optimization approach used (heuristic, policy classes or exact algorithms). Finally, we pay attention to the applications of the models.literature review;economic dependence;failure interaction;maintenance policies;grouping maintenance;multi-component systems;opportunistic maintenance;maintencance optimization;structural dependence

The Sugawara generators at arbitrary level

We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel--Kac--Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF ``oscillators'' unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac--Moody algebras, and $E_{10}$ in particular.Comment: 17 pages, 1 figure, LaTeX2e, amsfonts, amssymb, xspace, PiCTe

Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example.non-stationary gamma process;condition-based maintenance;degradation modelling;imperfect maintenance;life-cycle management;renewal-type dynamic programming equation

Monodromy Matrix in the PP-Wave Limit

We construct the monodromy matrix for a class of gauged WZWN models in the plane wave limit and discuss various properties of such systems.Comment: 16 page

Missing Modules, the Gnome Lie Algebra, and E10E_{10}

We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK

Non-Standard Fermion Propagators from Conformal Field Theory

It is shown that Weyl spinors in 4D Minkowski space are composed of primary fields of half-integer conformal weights. This yields representations of fermionic 2-point functions in terms of correlators of primary fields with a factorized transformation behavior under the Lorentz group. I employ this observation to determine the general structure of the corresponding Lorentz covariant correlators by methods similar to the methods employed in conformal field theory to determine 2- and 3-point functions of primary fields. In particular, the chiral symmetry breaking terms resemble fermionic 2-point functions of 2D CFT up to a function of the product of momenta. The construction also permits for the formulation of covariant meromorphy constraints on spinors in 3+1 dimensions.Comment: 15 pages, Latex, LMU-TPW 94-1

E10 for beginners

Invited talk presented by H. Nicolai at the Feza Gürsey Memorial Conference, Istanbul, June 1994