A brief overview of strings propagating on noncompact coset spaces G/H is presented in terms of gauged WZW models. The role played by isometries in the existence of target space duality and by fixed points of the gauged transformations in the existence of singularities and horizons, is emphasized. A general classification of the spaces with a single time-like coordinate is presented. The spacet...

We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: In fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: If a,b are enumeration degrees, with a total, and a < b, then in the degree interval (a,b), one can embed every countable partial order.

We complete a study of the splitting/non-splitting properties of the enumeration degrees below 0′e by proving an analog of Harrington’s non-splitting theorem for the Σ2 enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.

We survey recent results on the enumeration of formal languages. In particular, we consider enumeration of regular languages accepted by deterministic and nondeterministic finite automata with n states, regular languages generated by regular expressions of a fixed length, and ω-regular languages accepted by Müller automata. We also survey the uncomputability of enumeration of context-free langu...

We show that every splitting of 0e in the local structure of the enumeration degrees, Ge, contains at least one low-cuppable member. We apply this new structural property to show that the classes of all K-pairs in Ge, all downwards properly Σ2 enumeration degrees and all upwards properly Σ2 enumeration degrees are first order definable in Ge.

In this paper we will give a formal description of succession rules in terms of linear operators satisfying certain conditions. This representation allows us to introduce a system of well-deened operations into the set of succession rules and then to tackle problems of combinato-rial enumeration simply by using operators instead of generating functions. Finally we will suggest several open prob...

A Costas array of order n is an n× n permutation matrix such that all vectors between pairs of ones are distinct. Thus, a permutation matrix fails to be a Costas array if and only if it contains ones that form a (possibly degenerate) parallelogram. In this paper, we enumerate parallelograms in an n × n permutation matrix. We use our new formulas to improve Davies’s O(n−1) result for the density...

For any k ≥ 0, all primes n satisfy the congruence nσk(n) ≡ 2 mod φ(n). We show that this congruence in fact characterizes the primes, in the sense that it is satisfied by only finitely many composite n. This characterization generalizes the results of Lescot and Subbarao for the cases k = 0 and k = 1. For 0 ≤ k ≤ 14, we enumerate the composite n satisfying the congruence. We also prove that an...

We enumerate the nonisomorphic and the distinct one-factorizations of K12. We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers.

This note presents a proof of Pólya’s random walk theorem using classical methods from special function theory and asymptotic analysis.