Closure theorem

Equn. (3.9) is a special case of the representation theorem or closure theorem, which is one of two important theorems that are used frequently in formal quantum mechanics. 

The energy denominators in the resonant and non-resonant parts of the polaris-ability [Eq.(2)] are each large and insensitive to the vibrational quantum numbers, Vg, of the intermediate electronic state e>. Since the Vg represent a complete orthonormal set of states, the sums over them in Eq.(6) may be evaluated by using the closure theorem, provided that the v-dependence of the energy denominators is negligible. The closure theorem may be stated essentiaUy as vXv = 1, and arises... 

If it is assumed that the difference in energy between the mean, p, of all levels of perturbing configurations and that of any level i of the f configuration, (PX is very large such that p — ,(P) is effectively constant for all states i, then the closure theorem is valid and the effects of configuration interaction can be represented [23] by... 

Abstract Fundamentals of amplitude interferometry are given, complementing animated text and figures available on the web. Concepts as the degree of coherence of a source are introduced, and the theorem of van Cittert - Zemike is explained. Responses of an interferometer to a spatially extended source and to a spectrally extended one are described. Then the main methods to combine the beams from the telescopes are discussed, as well as the observable parameters - vibilities and phase closures. 

The second statement of Theorem 3.1 follows from the following fact the closure of a G -orbit is a union of orbits of smaller dimensions. Hence any orbit contains a closed G -orbit in its closure. (Moreover, it is unique by Theorem 3.3.)... 

Theorem 3.6 (Birkes [8], Kempf [44]). If x e V and Y is a nonempty closed invariant subset contained in the closure of x, then there exists a one-parameter subgroup A C G such that limt o A(t)x = y for some y eY. 

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

The part 5qo can be defined as the image of /"1( oonX°rd) under 7r or as the locus where Go cz pp and G pp. Similar for 5oi, 5j0 and Sn. The above arguments show that 5(2,p) x Spec(Fp) is the union of the closures of the Sij. Hence if we show that Sij is irreducible then these closures are the components of 5(2,p) x Spec(Fp). A standard argument using the theorem on the monodromy of the p-torsion points of the universal abelian scheme over A2 X Spec(Fp) (see [Ek] or [FC]) shows that 5 j is indeed irreducible. 

This table is derived as follows. We have I A =AI = A, and 11 = I. The only remaining product is AA, which by closure must be either A or /. If A A were equal to A, then A would occur twice in column 2 (and twice in row 2), thereby violating the theorem proved above. 

Theorem 6. If, in for some TzeK there exists a whole (x, -motion for which a(x, %) does not lie entirely in the closure of a>T(fc) [oc(x, k) coT(k)], then there exist rj2 slow relaxations. 

Answers to the following questions are sought. (1) Can a closure, or several closures, be found to satisfy theorems for simple liquids (2) If such closures exist, will they be improvements over conventional ones Do they give better thermodynamic and structural information (3) Will such closure relations render the IE method more competitive with respect of computer simulations and with other methods of investigation ... 

This closure contains three parameters that will have to be determined thanks to consistency conditions and theorems (the Zero-Separation Theorems) connecting the properties of the bulk fluid with the correlation functions at coincidence (r = 0). This last relation reduces to (38) if ( and numerical implementation of consistency conditions with more than one parameter in the context of an iterational theory is far from being trivial. 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

We have mentioned already that when the MSA or HNC closure is used with Eq. (71), the contact value of the density profile is some function of the inverse compressibility, dp/dp rather than the pressure recall Eq. (87). In the OZ2 theory, Eq. (87) is satisfied formally if the BGY equation is used. If the LMBW equation is used, there is no known exact theorem. However, experience has shown that Eq. (87) seems to be satisfied closely. 

When nanotubes are synthesized,8,26,27 they are typically closed (capped) at both ends. For the (9,0) and the (5,5), the caps can be fullerene hemispheres 6 in general, however, caps can have various structures, shapes and degrees of curvature.8 One requirement that they do have to satisfy is imposed by Euler s theorem, 8,26,27 according to which the closure of any hexagonal framework can be achieved only by the introduction of exactly twelve pentagons. Thus, each cap must have six. 

The spin term is straightforward to evaluate by the Wigner-Eckart theorem but the rotational term requires further consideration. Let us introduce the projection operator onto the complete set of rotational functions between the operators Pp (J) and 2) (< >) (the closure relationship) ... 

Arbitrary groups of matrices are not our main concern, but we should record some simple relations between such groups and their closures. Apart from allowing more general statements of some later theorems, this will be useful because extension to a larger field involves taking-closures. 

Theorem. Let k be an infinite field. The closed subsets of k , with polynomial maps, are precisely equivalent to certain representable functors. The equivalence preserves products, and rakes closed subsets to closed subfunctors represented by quotient rings). Closure in a larger C corresponds to base extension. 

Theorem. Let k and k, be the algebraic and separable closures of k. Let A be a finite-dimensional k-algebra. The following are equivalent ... 

Proof. The closure of N over k is still nilpotent, and by (9.2) the decomposition of elements takes place in k, so we may assume k is algebraically closed. The center of N is an abelian algebraic matrix group to which (9.3) applies. If the set Ns is contained in the center, it will then be a closed subgroup, and the rest is obvious from the last theorem. Thus we just need to show Nt is central. 

Consider connected closed subgroups H of G which are normal and solvable. If Ht and H2 are such, so is (the closure of) H H, since the dimensions cannot increase forever, there is actually a largest such subgroup. We denote it by R and call it the radical of G. By (10.3), the unipotent elements in R form a normal subgroup U, the unipotent radical. We call G semisimple if R is trivial, reductive if U is trivial. The theorem then (for char(fc) = 0) is that all representations are sums of irreducibles iff G is reductive. It is not hard to see this condition implies G reductive (cf. Ex. 20) the converse is the hard part. We of course know the result for R, since by (10.3) it is a torus we also know that this R is central (7.7), which implies that the R-eigenspaces in a representation are G-invariant. The heart of the result then is the semisimple case. This can for instance be deduced from the corresponding result on Lie algebras. 

Theorem, (a) Let 0 R be a finitely generated algebra over a field k. Then R has a k-algebra homomorphism to the algebraic closure ... 

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