The wonderland of reflections

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable \omega-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.Comment: 24 page

The ERA of FOLE: Superstructure

This paper discusses the representation of ontologies in the first-order logical environment FOLE (Kent 2013). An ontology defines the primitives with which to model the knowledge resources for a community of discourse (Gruber 2009). These primitives, consisting of classes, relationships and properties, are represented by the ERA (entity-relationship-attribute) data model (Chen 1976). An ontology uses formal axioms to constrain the interpretation of these primitives. In short, an ontology specifies a logical theory. This paper is the second in a series of three papers that provide a rigorous mathematical representation for the ERA data model in particular, and ontologies in general, within the first-order logical environment FOLE. The first two papers show how FOLE represents the formalism and semantics of (many-sorted) first-order logic in a classification form corresponding to ideas discussed in the Information Flow Framework (IFF). In particular, the first paper (Kent 2015) provided a "foundation" that connected elements of the ERA data model with components of the first-order logical environment FOLE, and this second paper provides a "superstructure" that extends FOLE to the formalisms of first-order logic. The third paper will define an "interpretation" of FOLE in terms of the transformational passage, first described in (Kent 2013), from the classification form of first-order logic to an equivalent interpretation form, thereby defining the formalism and semantics of first-order logical/relational database systems (Kent 2011). The FOLE representation follows a conceptual structures approach, that is completely compatible with Formal Concept Analysis (Ganter and Wille 1999) and Information Flow (Barwise and Seligman 1997)

Reciprocity constraints on the matrix of reflection from optically anisotropic surfaces

We derive certain constraints on the reflection matrix for reflection from a plane, nonmagnetic, optically anisotropic surface using a reciprocity theorem stated long ago by van de Hulst in the context of scattering of polarized light. The constraints are valid for absorbing and chiral media and can be used as tools to check the consistency of derived expressions for such matrices in terms of the intrinsic parameters of the reflecting medium as illustrated by several examples.Comment: 14 pages, 2 figures, submitted to Jour. Opt. Soc. Am.

Boundary conditions for coupled quasilinear wave equations with application to isolated systems

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on $\partial\Sigma$. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell's equations in the Lorentz gauge and Einstein's gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.Comment: 22 pages, no figure