L3

We provide both a diagrammatic and logical system to reason about

quantum phenomena. Essential features are entanglement, the flow of

information from the quantum systems into the classical measurement

contexts, and back---these flows are crucial for several quantum informatic

scheme's such as quantum teleportation---, and mutually unbiassed

observables---e.g. position and momentum. The formal structures we use are

kin to those of topological quantum field theories---e.g. monoidal

categories, compact closure, Frobenius objects, coalgebras. We show that

our diagrammatic/logical language is universal. Informal

appetisers can be found in:

* Introducing Categories to the Practicing Physicist

http://web.comlab.ox.ac.uk/oucl/work/bob.coecke/Cats.pdf

* Kindergarten Quantum Mechanics

http://arxiv.org/abs/quant-ph/0510032

We show that the profinite completion (a universal algebraic

construction) and the MacNeille completion (an order-theoretic

construction) of a modal algebra $A$ coincide, precisely when the congruences of finite index of $A$ correspond to principal order filters. Examples of such modal algebras are the free K4-algebra and the free PDL-algebra on finitely many generators.

Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. The obtained multiscale approximations are computationally cheap and robust. Some of results obtained are rather surprising and unexpected. First of all is the zero-one asymptotic of eigenvectors (asymptotically exact lumping; but these asymptotic lumps could intersect). Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery"

for reaction graphs. Roughly speaking, the dynamics of linear multiscale networks transforms into the dynamics on finite sets of reagent names.

A.N. Gorban and O. Radulescu, Dynamic and static limitation in multiscale reaction networks, Advances in Chemical Engineering, 2008 (in press), arXiv e-print: http://arxiv.org/abs/physics/0703278

A.N. Gorban and O. Radulescu, Dynamical robustness of biological networks with hierarchical distribution of time scales, IET Syst.

Biol., 2007, 1, (4), pp. 238-246.