These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

Generalising Nachbin's theory of ``topology and order'', in this paper we   continue the study of quantale-enriched categories equipped with a compact   Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces   with ultrafilter-quantale-enriched categories, and show that the presence of a   compact Hausdorff topology guarantees Cauchy completeness and (suitably   defined) ...

Background: The prevalence of reported allergic diseases is higher in North-America than in other countries around the world. Further, certain types of allergies are more common in some geographic regions or amongst different ethnic populations. This phenomenon has not been well studied amongst the Persian population in the Canadian context; therefore, the current study aimed to compare the pre...

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The descriptions some systems, such as classical mechanical require that do not have pullbacks, and this limits the utility span a formal framework. Given $\mathscr{C}$ $\mathscr{C}^\prime$ functor $\mathcal F$ from to $\mathscr{C}^\prime$, we introduce notion pullback cospan in $\mathscr{C}$, ...

Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind enriched category. The base for enrichment is category commutative monoids—or in straightforward generalisation, modules over rig k . However, tensor product on this not usual one, but rather warping it by certain monoidal comonad Q Thus sense, skew sense Szlachányi. Our first main result ...