Proof of the formula

Let X be a quasi-projective nonsingular surface and the Hilbert scheme of n-points on X. 

Lemma 6.9. k is finite and birational. It induces an isomorphism between 

Moreover, it is clear that vrl -qs nx) 7t (S X) S X is a topological fiber bundle because the corresponding assertion holds for tt sr ,v = A. 

But we know that the fiber 7t (C ) is irreducible. Hence W, / is nontrivial only when 0 is trivial. In other words, we have K = Wtuviai- We also have 

On the other hand, since S X has only quotient singularities by finite groups, we have 

Lemma 6.10. (1) The Hilbert-Chow morphism 7r X SnX is semismall with respect to the stratification SnX = JUS X (1.6). 

---

This chapter gives a brief summary of our new approach without showing detailed proofs of the formulae presented here. However, when we use the formulae, we indicate the original papers where one can find their proofs. Here we concentrate on the qualitative aspects. 

Aa thare e no proof of the formula aaaigned to thle flrat product.lt may be H-00-CH(0 Ha )-OOOH,which... 

Remark In the proof of the equality ind 0 + ind G for semisimple Lie algebras, we may use the Rais es formula [205] for the index of a semidirect product. But in this case, as before, the Lie algebra is to be found, which makes up the crucial point in the proof of the formula for the index of the Lie algebra G. The theorem follows. 

The experimental proof of the formula is possible from several directions. 

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

The proof of this formula is omitted here. It should be noted that from such reasoning it seems clear that due to the summarized approximation in the space Hg-i the convergence occurs in the space Hb- That is to say, the conditions... 

Another proof of the configuration of D-mannitol and also of D-manno-n-manno-octitol (XVI), which is likewise dependent on the experimental proof of the equivalent symmetry of D-mannitol is the following. D-Mannose has been converted, by successive cyanohydrin syntheses, first to a mannoheptose and then to a mannooctose which on reduction yielded a mannooctitol whose octaacetyl derivative is optically inactive. (It was not possible to examine the octitol itself because of its very low solubility in water.)87 The meso character of the octaacetate shows that the mannooctitol must possess a meso configuration, with a plane of symmetry between carbon atoms 4 and 5. To write its formula, the hydroxyl at carbon atom 7 is placed on the... 

The formula of 3,5-diamino-oxadiazole has been attributed to the product resulting from the action of hydroxylamine on dicyandiamide 88) (see p. 826). No proof of the proposed structure was given but in the light of some novel work 18a) it appears to be extremely doubtful. In fact, it could be shown that the product forms a monohydrochloride and reacts with only one mole of nitrous acid therefore the structure of a hydroxyaminotriazole (L) which is isomeric with the proposed formula cannot be ruled out. 

In this section we give a proof of the Kawabata formula (52), following a method due to Kaveh (1984) and Mott and Kaveh (1985a, b). We assume that an electron undergoes a random walk, which determines an elastic mean free path l and diffusion coefficient D. If an electron starts at time t=0 at the point r0 then the probability per unit volume of finding it at a distance r, at time U denoted by n(r, t) obeys a diffusion equation... 

It is possible to prepare the hydrosulphites of the metals in anhydrous condition by the action of various metals, for instance zinc, magnesium, sodium, on sulphur dioxide in the presence of moisture-free ether 4 or alcohol.5 This reaction is of importance as having supplied an early proof of the absence of hydrogen from the salts and hence of the incorrectness of the formula Zn(HSOa)a, NaHS02, which had been accepted previously ... 

The constitution of many of the heteropoly-acids and of their salts may thus be explained, although direct proof of the correctness of some of these formulae is lacking. Some of the heteropoly-compounds of vanadium are, however, very complex, and cannot be represented by application of the foregoing scheme. To take an extreme instance, Rogers1 prepared a black, crystalline compound to which he ascribed the following formula —... 

Additional proof of the proposed formula for (—)-quebrachitol was furnished by the synthesis of 1 -O-methy 1-Zeyo-inositol (2-O-methyl-feao-inositoI)... 

It should be mentioned that in some cases it may happen that a questioned ink can be more positively identified through presence of fluorescent or other unique components in the formulation. When sufficient questioned ink is available and the proprietory formula composition has been furnished, further analysis can lead to the identification of a component which may provide additional proof of the identity of the ink. For example, there are a variety of fatty acids, resins, and viscosity adjusters added to inks which can be readily identified by TLC or gas liquid chromatography (GLC), when sufficient ink is available. As further examples, amorphous carbon and graphite, which are common dispersion ingredients in ballpoint inks, can be distinguished using electron diffraction methods. 

---