R. Penrose, Techniques of Differential Topology in Relativity, CBMS Regional Conference Series in Applied Mathematics, No. 7., SIAM, Philadelphia, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of dipaths and observe their evolution in time. In fact, this is fairly close to methods used in fault tolerant distributed systems theory [33] Definition 12. Let (U; 6) be a partially ordered set A subset V ae U is called achronal if for all x; y 2 V : x 6 y ) x = y (similarly to the notion in [48]) Definition 13. Let (X; 6) be a po space. 1. X; 6) is a parameterized po space if there exists a dimap F : X IR such that X t : F Gamma1 (t) be achronal for all t 2 IR. 2. 6 is Euclidian, if there exists a finite number of dimaps f i : X IR such that 8x; y 2 X : x y , 8i : f i (x) ....

Penrose, R., "Techniques of Differential Topology in Relativity," Conference Board of the Mathematical Sciences, Regional Conference Series in Applied Ma thematics 7, SIAM, Philadelphia, USA, 1972.

....are given, but we will only see in Section 6 that cubical complexes (or Higher Dimensional Automata, 27] and [49] give rise naturally to such spaces, hence most combinatorial concurrency models are instances of these local po spaces. It is worth noting that some models in General Relativity [48] consider timed spaces, and the authors benefited from some of these physical concepts when developing this theory. Section 4 then gives the first definitions of the new homotopy theory we need in order to define equivalence of paths along the intuitions developped in Section 2. A central notion ....

....(0) Their concatenation f 1 f 2 is again a dipath. f 1 f 2 ) t) ae f 1 (t) t 0:5; f 2 (2t Gamma 1) t 0:5: 2. One might look at maps from arbitrary intervals and allow equivalence classes with respect to strictly increasing homeomorphisms between intervals. Definition 3.8. Compare [48]. Let X be a locally partially ordered space. We define a new relation OE on X by x OE y if there is a dipath from x to y in X. Lemma 3.9. If X has a global partial order, the relation OE is a new partial order. Proof. The relation OE is coarser then the relation , i.e. x OE y ) x y. Hence, ....

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R. Penrose, Techniques of Differential Topology in Relativity, Conference Board of the Mathematical Sciences, Regional Conference Series in Applied Mathematics, vol. 7, SIAM, Philadelphia, USA, 1972.

....fundamental group, universal covering, holonomy, and fundamental polyhedron. All these concepts have very formal and abstract definitions that can be found in classical textbooks in topology (for instance, 106, 116] and, in the particular context of Lorentzian manifolds used in relativity, [121, 66]) In this primer we just provide pictorial definitions with no lack of rigour, we hope illustrated mostly in the cases of locally Euclidean surfaces. The strategy for characterizing spaces is to produce invariants which capture the key features of the topology and uniquely specify each ....

Penrose, R. 1972, Techniques of Differential Topology in Relativity, Society for Industrial and Applied Mathematics, Philadelphia

....are given, but we will only see in Section 6 that cubical complexes (or Higher Dimensional Automata, 28] and [50] give rise naturally to such spaces, hence most combinatorial concurrency models are instances of these local po spaces. It is worth noting that some models in General Relativity [49] consider timed spaces, and the authors benefited from some of these physical concepts when developing this theory. Section 4 then gives the first definitions of the new homotopy theory we need in order to define equivalence of paths along the intuitions developped in Section 2. A central notion ....

....(0) Their concatenation f 1 f 2 is again a dipath. f 1 f 2 ) t) ae f 1 (t) t 0:5; f 2 (2t Gamma 1) t 0:5: 2. One might look at maps from arbitrary intervals and allow equivalence classes with respect to strictly increasing homeomorphisms between intervals. Definition 3.9. Compare [49]. Let X be a locally partially ordered space. We define a new relation OE on X by x OE y if there is a dipath from x to y in X. Lemma 3.10. If X has a global partial order, the relation OE is a new partial order. Proof. The relation OE is coarser then the relation , i.e. x OE y ) x y. Hence, ....

[Article contains additional citation context not shown here]

R. Penrose, Techniques of Differential Topology in Relativity, Conference Board of the Mathematical Sciences, Regional Conference Series in Applied Mathematics, vol. 7, SIAM, Philadelphia, USA, 1972.

....in Sect. I, any hypersurface Sigma ae M is an initial value surface if its lift to M is a Cauchy surface. A local algebra of observables can be defined on any neighborhood U ae M for which (i) U; gj U , regarded as a spacetime, is globally hyperbolic, and (ii) U is connected and causally convex [15]. An open set U is causally convex if no causal curve in M intersects U in a disconnected set. If U is not causally convex, then some points that are spacelike separated in the spacetime U; gj U are joined by a null or timelike curve in M , and the commutation relations for field operators can ....

....related by Eq. 2. 28) Although a consistent definition of a collection of states will require an additional unwanted restriction on the size of neighborhoods, the structure of local algebras coincides 2 Although conditions (i) and (ii) above resemble what is called the local causality property [15], the latter is much more restrictive: the closure of a local causality neighborhood is required to lie in a geodesically convex normal neighborhood. with that of the globally hyperbolic spacetime. Let C = f(U; O) U; O) g be the collection of all pairs with U 2 C and O a choice of ....

R. Penrose, Techniques of Differential Topology in Relativity, Soc. for Industrial and Applied Math., Philadelphia, 1972.

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R. Penrose, Techniques of Differential Topology in Relativity, CBMS Regional Conference Series in Applied Mathematics, No. 7., SIAM, Philadelphia, 1972.

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R. Penrose, Techniques of differential topology in relativity, SIAM, Philadelphia, 1972, (Regional Conf. Series in Appl. Math., vol. 7).

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R. Penrose, Techniques of differential topology in relativity, SIAM, Philadelphia, 1972, (Regional Conf. Series in Appl. Math., vol. 7). 71

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