Verallgemeinerung eines Satzes von F. Riesz

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: XR, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.The purpose of this paper is to represent such a bounded linear operator A: C(X)F as an integral Af= d with respect to some content , defined on the algebraa(X) which is generated by the open sets UX.This representation is unique, if is required to be regular (p. 11).     

Algebraic aspects of crystallography

Taking into account the fact that space groups are groups of transformations of Euclideann-dimensional space, non-equivalent systems of non-primitive translations are defined. They can be brought into one-to-one correspondence with the elements of the groupH ¹ (K, R ⁿ /Z ⁿ ) or with those of the groupH ¹ (K, Z ⁿ /kZ ⁿ )/H ¹ (K, Z ⁿ ). (K is a point group of orderk.) The consistency of these findings with the results of Part I is given by the isomorphisms $$H^2 (K,Z^n ) \cong H^1 (K,R^n /Z^n ) \cong H^1 (K,Z^n /kZ^n )/H^1 (K,Z^n ).$$ Theorems are proved giving the conditions for cohomology groupsH ^q (K, A) to be zero. These conditions are fulfilled in particular ifA=R ⁿ andK is a subgroup ofGL (n, R) that either is compact (thenq>0) or has a finite normal subgroup leaving no element ofR ⁿ invariant (thenq≧0). This implies that the affine, the Euclidean and the inhomogeneous Lorentz groups are the only extensions ofR ⁿ by the corresponding homogeneous groups. By way of illustration, the theory of this paper is applied to two 2-dimensional space groups.     

The electronic spectra of acenaphthylene and fluoranthene

Acenaphthylene (I) and fluoranthene (II) in their electronic ground state are often regarded as consisting of two weakly interacting subsystems (naphthylene/ethylene or naphthalene/ benzene), which conserve much of their individuality. This seems to suggest that the lower electronically excited states of I and II can also be linked in a simple fashion to the low energy L _b , L _a , B _b , and B _a states of naphthalene and of benzene in II and of naphthalene in I, according to either Simpson's Independent System Approach or the Molecules in Molecules scheme proposed by Longuet-Higgins and Murrell. Contrary to expectation this approach fails even for the lowest excited states of I and II, because of the extensive mixing of high energy locally excited and charge transfer configurations. These involve molecular orbitals of the subsystems naphthalene/ethylene or naphthalene/benzene, which do not contribute to their low energy L _b , L _a , B _b or B _a states.

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Zusammenfassung Es wird oft angenommen, daß sich Acenaphthylen (I) und Fluoranthen (II) in ihrem elektronischen Grundzustand in guter Näherung so verhalten, wie wenn sie sich aus zwei nur locker miteinander gekoppelten Teilsystemen (Naphthalin/Aethylen, bzw. Naphthalin/ Benzol) zusammensetzen würden. Dies legt zunächst die Vermutung nahe, daß auch die niedrigst liegenden elektronisch angeregten Zustände von I und II in einfacher Weise mit den L _b , L _a , B _b und B _a Zuständen des Naphthalins und des Benzols in Beziehung setzen lassen, indem man sich des von Simpson vorgeschlagenen Independent System Approach oder des Molecules in Molecules Verfahrens von Longuet-Higgins und Murrell bedient. Es zeigt sich aber, daß dies nicht der Fall ist.Der Grund für das Versagen der genannten Betrachtungsweise ist darin zu suchen, daß sowohl in I als auch in II bereits in den elektronisch angeregten Zuständen niedrigster Energie lokal angeregte Konfigurationen der Teilsysteme eine Rolle spielen, die nicht zu deren L _b , L _a , B _b und B _a Zuständen beitragen, sondern zu Zuständen höherer Energie.

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Résumé On suppose souvent que dans leurs états électroniques fondamentaux, l'acénaphtylène (I) et le fluoranthène (II) se comportent pratiquement comme s'ils étaient composés de deux systèmes partiels faiblement couplés (naphtalène/éthylène, respectivement naphtalène/benzène).On peut alors croire que les premiers états électroniques excités de I et de II peuvent être simplement mis en rapport avec les états L _b , L _a , B _b et B _a du naphthalène et du benzène, par exemple en se servant de la méthode «Independent System Approach » proposée par Simpson ou de la méthode «Molecules in Molecules» proposée par Longuet-Higgins et Murrell.Contrairement à cette attente, l'approximation mentionnée n'est plus valable, même pour les états excités de plus basse énergie. Ceci est dû au fait que des configurations d'excitation locale et de transfert de charge qui contribuent à ces états, et qui ont une énergie élevée, sont basées sur des orbitales moléculaires des systèmes partiels — naphtalène/éthylène ou naphtalène/benzène — qui ne contribuent pas aux états d'excitation locale de basse énergie, L _b , L _a , B _b ou B _a , de chacun de ces systèmes.

     

Averaged norms

A method to construct an equivalent norm with both a rotundity and a smoothness property in a Banach space having two different equivalent norms, one with the rotundity and one with the smoothness property.     

Ueber Elektrolyse

Ohne Zusammenfassung