The categorical theory of relations and quantizations

In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of $A-A$ bicomodules. Properties of relations are defined in terms of the symmetric monoidal structure. Equivalence relations are shown to be commutative monoids in the category of relations. Quantization in our view is a property of functors between monoidal categories. This notion of quantization induce a deformation of all algebraic structures in the category, in particular the ones defining properties of relations like transitivity and symmetry.Comment: corrected typo

Theory of linear G-difference equations

We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations,symmetries of an equation are module endomorphisms and conserved structures are invariants in the tensor algebra of the given equation. We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underluing set. We relate our notion of difference equations and solutions to systems of classical difference equations and their solutions and show that our notions include these as a special case.Comment: 34 page

Constructing a partially transparent computational boundary for UPPE using leaky modes

In this paper we introduce a method for creating a transparent computational boundary for the simulation of unidirectional propagation of optical beams and pulses using leaky modes. The key element of the method is the introduction of an artificial-index material outside a chosen computational domain and utilization of the quasi-normal modes associated with such artificial structure. The method is tested on the free space propagation of TE electromagnetic waves. By choosing the material to have appropriate optical properties one can greatly reduce the reflection at the computational boundary. In contrast to the well-known approach based on a perfectly matched layer, our method is especially well suited for spectral propagators.Comment: 32 pages, 19 figure

Copepod reaction to odor stimuli influenced by cestode infection

The cestode Schistocephalus solidus uses copepods as first and sticklebacks as second intermediate hosts. For transmission, an infected copepod has to be preyed upon by a stickleback. We used copepods of the species Macrocy albidus to test whether infected and uninfected copepods differ in their reaction to two kind of simultaneously presented odors: odors of sticklebacks and odors of sticklebacks and conspecificz. By giving this choice, we attempted to force the copepods to make a trade-off between the benefit of risk dilution and possible predator confusion and the costs of food competition and other disadvantages induced by conspecifics. Within 1-8 h after last feeding, uninfected copepods clearly preferred the odors of conspeciflcs under the chemically simulated threat of predation. This was in contrast to the infected copepods, who tended to avoid the odor of conspecifics. When the time between experiment and last feeding varied, infected copepods showed an increas preference for fish water only (or avoided conspecthcs) with increasing hunger level This suggests that S. solidus benefits from hunger-induced behavioral changes of its copepod host by influencing its microhabitat selection. The same effect could be found in both sexes; however, it was significantly more pronounced in male than in female copepods. We propose several hypotheses that could explain the difference between the sexes in their infection-dependent microhabitat selectio