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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
-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 , where is
a compact manifold with smooth boundaries . 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 . 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
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