On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

Efficient Groundness Analysis in Prolog

Boolean functions can be used to express the groundness of, and trace grounding dependencies between, program variables in (constraint) logic programs. In this paper, a variety of issues pertaining to the efficient Prolog implementation of groundness analysis are investigated, focusing on the domain of definite Boolean functions, Def. The systematic design of the representation of an abstract domain is discussed in relation to its impact on the algorithmic complexity of the domain operations; the most frequently called operations should be the most lightweight. This methodology is applied to Def, resulting in a new representation, together with new algorithms for its domain operations utilising previously unexploited properties of Def -- for instance, quadratic-time entailment checking. The iteration strategy driving the analysis is also discussed and a simple, but very effective, optimisation of induced magic is described. The analysis can be implemented straightforwardly in Prolog and the use of a non-ground representation results in an efficient, scalable tool which does not require widening to be invoked, even on the largest benchmarks. An extensive experimental evaluation is givenComment: 31 pages To appear in Theory and Practice of Logic Programmin

Irrational Conformal Field Theory

This is a review of irrational conformal field theory, which includes rational conformal field theory as a small subspace. Central topics of the review include the Virasoro master equation, its solutions and the dynamics of irrational conformal field theory. Discussion of the dynamics includes the generalized Knizhnik-Zamolodchikov equations on the sphere, the corresponding heat-like systems on the torus and the generic world- sheet action of irrational conformal field theory.Comment: 195 pages, Latex, 12 figures, to appear in Physics Reports. Typos corrected in Sections 13 and 14, and a footnote added in Section 1

Inside-Out Polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart's theory of lattice-point counting to a convex polytope dissected by a hyperplane arrangement. We particularly develop the applications to graph and signed-graph coloring, compositions of an integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat

Faces of Birkhoff Polytopes

The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face with combinatorial type L. By a result of Billera and Sarangarajan, a combinatorial type L of a d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is at most d. We will characterize those types whose Birkhoff dimension is at least 2d-3, and we prove that any type whose Birkhoff dimension is at least d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for d between 2 and 8.Comment: 29 page