By Marcelo P Fiore; Cambridge University Press

ISBN-10: 0511526563

ISBN-13: 9780511526565

ISBN-10: 0521602777

ISBN-13: 9780521602778

Axiomatic specific area conception is important for realizing the which means of courses and reasoning approximately them. This e-book is the 1st systematic account of the topic and experiences mathematical buildings appropriate for modelling sensible programming languages in an axiomatic (i.e. summary) environment. particularly, the writer develops theories of partiality and recursive forms and applies them to the examine of the metalanguage FPC; for instance, enriched specific versions of the FPC are outlined. in addition, FPC is taken into account as a programming language with a call-by-value operational semantics and a denotational semantics outlined on best of a express version. To finish, for an axiomatisation of absolute non-trivial domain-theoretic types of FPC, operational and denotational semantics are comparable by way of computational soundness and adequacy effects. To make the publication quite self-contained, the writer contains an advent to enriched type conception

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**Extra info for Axiomatic domain theory in categories of partial maps**

**Example text**

Since, on experimental grounds, the initial state of an orbit is never known accurately, we cannot "predict" where the system will be at some later time. To explain unpredictability, we introduce the definition of stable and unstable orbits. The presence of unstable orbits plays an important role in dynamical systems. Let us consider the one-dimensional difference equation x{t + \)=f{x{t)\ ? = 0,l,--. 1. 1) is locally stable if for every s > 0 there exists S > 0 such that I < 5 implies that llx, - x* < s for all f > 1.

1). 1) is a sequence {(*,}7=o t n a t satisfies the equation for all t = 0,1, . 1) so that the solution satisfies the initial condition is called the initial value problem. 1) for all t = 0 , 1 , a n d involves a 22 2. SCALAR LINEAR DIFFERENCE EQUATIONS constant C that can be determined once an initial value is prescribed. 1) for all t = 0, 1, . 1. The sequence is denoted by O(x0) and is called the orbit or trajectory of the system starting from Xs. 2. 1) if x=f(x). 2) or as a solution to the system of equation * = /(*) We also call x afixed (or stationary or equilibrium) point of f.

Hence, the equilibrium point is a source. Example Consider x{t + l)=-x'{t)+x(t), f=0,l,-. The difference equation has a unique equilibrium point x* = 0. As 1, / ' ( 0 ) = 0, /-(0) = - 6 < 0 , we conclude that x* is asymptotically stable. In the case of / ' [x ) = - 1, the map / is not monotone but rather oscillatory, and it flips from a point close to x* to the other side of x*. If the equilibrium point x" becomes unstable, and orbit cannot approach x*. But if the iterates remain bounded, it is possible that the odd iterates converge to a limit point p, and the even iterates converge to a limit point f(p).

### Axiomatic domain theory in categories of partial maps by Marcelo P Fiore; Cambridge University Press

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