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Mesablishvili B.,Tbilisi Center for Mathematical science | Mesablishvili B.,Razmadze Mathematical Institute
Applied Categorical Structures | Year: 2013

In the theory of coalgebras C over a ring R, the rational functor relates the category C*M of modules over the algebra C* (with convolution product) with the category CM of comodules over C. This is based on the pairing of the algebra C* with the coalgebra C provided by the evaluation map ev:C*⊗R C R. The (rationality) condition under consideration ensures that CM becomes a coreflective full subcategory of C*M. We generalise this situation by defining a pairing between endofunctors T and G on any category A as a map, natural in a,b ∈ A, βa,b:A(a, G(b)) → A(T(a),b), and we call it rational if these all are injective. In case T = (T, m T, e T ) is a monad and G = (G, δ G, ε G ) is a comonad on A, additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ®, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules AT which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories. © 2011 Springer Science+Business Media B.V.

Bezhanishvili G.,New Mexico State University | Bezhanishvili N.,Imperial College London | Gabelaia D.,Razmadze Mathematical Institute | Kurz A.,University of Leicester
Mathematical Structures in Computer Science | Year: 2010

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces - the duals of Boolean algebras - and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality. Copyright © Cambridge University Press 2010.

Dorn H.,Humboldt University of Berlin | Drukker N.,Humboldt University of Berlin | Jorjadzea G.,Razmadze Mathematical Institute | Kalousios C.,Humboldt University of Berlin
Journal of High Energy Physics | Year: 2010

We present a four parameter family of classical string solutions in AdS 3 × S 3, which end along a light-like tetragon at the boundary of AdS3 and carry angular momentum along two cycles on the sphere. The string surfaces are space-like and their projections on AdS 3 × S 3 have constant mean curvature. The construction is based on the Pohlmeyer reduction of the related sigma model. After embedding in AdS 5 × S 5, we calculate the regularized area and analyze conserved charges. Comments on possible relations to scattering amplitudes are presented. We also sketch time-like versions of our solutions. © SISSA 2010.

Blankleider B.,Flinders University | Kvinikhidze A.N.,Razmadze Mathematical Institute | Skawronski T.,Flinders University
AIP Conference Proceedings | Year: 2010

A crossing symmetric πN scattering amplitude is constructed through a complete attachment of two external pions to the dressed nucleon propagator of an underlying πN potential model. Our formulation automatically provides expressions also for the crossing symmetric and gauge invariant pion photoproduction and Compton scattering amplitudes. We show that our amplitudes are unitary if they coincide on-shell with the amplitudes obtained by attaching one pion to the dressed πNN vertex of the same potential model. © 2010 American Institute of Physics.

Esakia L.,Razmadze Mathematical Institute
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2011

The modal system S4.Grz is the system that results when the axiom (Grz) □(□(p → □p) → p) → □p is added to the modal system S4, i. e. S4.Grz = S4 + Grz. The aim of the present note is to prove in a direct way, avoiding duality theory, that the modal system S4.Grz admits the following alternative definition: S4.Grz = S4 + R-Grz, where R-Grz is an additional inference rule: (R-Grz) ⊢ □(p → □p) → p/⊢ p This rule is a modal counterpart of the following topological condition: If a subset A of a topological space X coincides with its Hausdorff residue ρ(A) then A is empty. In other words the empty set is a unique "fixed" point of the residue operator ρ(·). We also present some consequences of this alternative axiomatic definition. © 2011 Springer-Verlag.

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